Stochastic Partial Di erential Equations: Analysis and ... · ysis of Stochastic Partial Di erential Equations" in the spring semester 2014 and in the spring semester 2015. These

Stochastic Partial Differential Equations:Analysis and Numerical Approximations

Arnulf Jentzen

May 23, 2016

2

Preface

These lecture notes have been written for the course “401-4606-00L Numerical Anal-ysis of Stochastic Partial Differential Equations” in the spring semester 2014 and inthe spring semester 2015. These lecture notes are far away from being complete andremain under construction. In particular, these lecture notes do not yet contain asuitable comparison of the presented material with existing results, arguments andnotions in the literature. This will be the subject of a future version of these lecturenotes. Furthermore, these lecture notes do not contain a number of proofs, argu-ments and intuitions. For most of this additional material, the reader is referredto the lectures of the course “401-4606-00L Numerical Analysis of Stochastic PartialDifferential Equations” in the spring semester 2014. Sonja Cox and Ryan Kurniawanare gratefully acknowledged for their very helpful advice and assistance, especiallyfor their help with the Matlab programs. Daniel Conus is also gratefully acknowl-edged for several comments that helped to improve the presentation of the results.In addition, we thank Antti Knowles and Benno Kuckuck for fruitful discussions.The students of the course “401-4606-00L Numerical Analysis of Stochastic PartialDifferential Equations” in the spring semester 2014 are gratefully acknowledged forpointing out a number of misprints to me. Special thanks are due to Timo Welti forbringing a number of misprints to my notice.

Zurich, March 2016

Arnulf Jentzen

3

Exercises

Solutions to the exercises can be turned in the designated mailbox in the anteroomHG G 53.x.

Exerc. Exercises Deadlinesheet1 Exerc. 1.1.8, 1.1.9, 1.1.10, & 2.2.6. 17.03.2016, 10:15 AM2 Exerc. 2.3.7, 2.4.20, 2.4.23, 3.4.23, 3.6.4, 3.6.5, & 3.6.18. 07.04.2016, 10:15 AM3 Exerc. 3.6.26, 4.3.3, 4.8.7, & 6.1.20 21.04.2016, 10:15 AM4 Exerc. 6.3.17 28.04.2016, 10:15 AM5 Exerc. 7.1.17 & 7.1.18 12.05.2016, 10:15 AM6 Exerc. 7.2.4 19.05.2016, 10:15 AM7 Exerc. 8.1.6 & 8.1.14 26.05.2016, 10:15 AM8 Exerc. 8.2.3 01.06.2016, 10:15 AM

4

Contents

I Foundations in mathematical analysis 13

1 Gronwall-type inequalities 151.1 Properties of the beta and the gamma function . . . . . . . . . . . . 15

1.1.1 Functional equation of the gamma function . . . . . . . . . . . 161.1.2 Monotonicity properties of the gamma and the beta function . 161.1.3 Estimates for the beta and the gamma function . . . . . . . . 17

1.2 Integral operators related to the beta function . . . . . . . . . . . . . 201.3 Generalized exponential-type functions . . . . . . . . . . . . . . . . . 221.4 Generalized time-continuous Gronwall-type inequalities . . . . . . . . 23

1.4.1 Gronwall-type inequalities with singularities at initial time . . 251.4.2 Gronwall-type inequalities without singularities at initial time 27

2 Nonlinear functions and nonlinear spaces 292.1 Sets and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.1 General functions . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.2 Preordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.1 Nonlinear characterization of the Borel sigma-algebra . . . . . 322.2.2 Pointwise limits of measurable functions . . . . . . . . . . . . 33

2.3 Strongly measurable functions . . . . . . . . . . . . . . . . . . . . . . 342.3.1 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.3 Strongly measurable functions . . . . . . . . . . . . . . . . . . 362.3.4 Pointwise approximations of strongly measurable functions . . 372.3.5 Sums of strongly measurable functions . . . . . . . . . . . . . 40

2.4 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 412.4.2 Semi-metric spaces . . . . . . . . . . . . . . . . . . . . . . . . 44

5

6 CONTENTS

2.4.3 Continuity properties of functions . . . . . . . . . . . . . . . . 452.4.4 Modulus of continuity . . . . . . . . . . . . . . . . . . . . . . 462.4.5 Extensions of uniformly continuous functions . . . . . . . . . . 48

3 Linear functions and linear spaces 513.1 Sums over possibly uncountable index sets . . . . . . . . . . . . . . . 51

3.1.1 Cofinal sequences . . . . . . . . . . . . . . . . . . . . . . . . . 513.1.2 Sums over possibly uncountable index sets . . . . . . . . . . . 543.1.3 Fubini for sums . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Sets of integrable functions . . . . . . . . . . . . . . . . . . . . . . . . 573.2.1 Lp-sets of measurable functions for p P r0,8q . . . . . . . . . . 573.2.2 Lp-spaces of strongly measurable functions for p P r0,8q . . . 58

3.3 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . 603.4.2 Best approximations and projections in Hilbert spaces . . . . 623.4.3 Examples of orthonormal bases . . . . . . . . . . . . . . . . . 63

3.4.3.1 Elementary properties of trigonometric functions . . 633.4.3.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq . . . . . . . . 653.4.3.3 Transformations of orthonormal bases . . . . . . . . 71

3.5 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5.1 Continuous linear functions on normed vector spaces . . . . . 733.5.2 Nuclear operators on Banach spaces . . . . . . . . . . . . . . . 75

3.5.2.1 Definition of Nuclear operators . . . . . . . . . . . . 753.5.2.2 Relation of bounded linear operators and nuclear op-

erators . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.2.3 Structure of the space of nuclear operators . . . . . . 773.5.2.4 Ideal property of the set of nuclear operators . . . . 783.5.2.5 Characterization of nuclear operators . . . . . . . . . 79

3.5.3 Hilbert-Schmidt operators on Hilbert spaces . . . . . . . . . . 793.5.3.1 Independence of the orthonormal basis . . . . . . . . 793.5.3.2 The Hilbert space of Hilbert-Schmidt operators . . . 803.5.3.3 Hilbert-Schmidt embeddings . . . . . . . . . . . . . . 81

3.6 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 823.6.1 Laplace operators on bounded domains . . . . . . . . . . . . . 84

3.6.1.1 Laplace operators with Dirichlet boundary conditions 843.6.1.2 Laplace operators with Neumann boundary conditions 863.6.1.3 Laplace operators with periodic boundary conditions 87

CONTENTS 7

3.6.2 Spectral decomposition for a diagonal linear operator . . . . . 88

3.6.3 Fractional powers of a diagonal linear operator . . . . . . . . . 92

3.6.4 Domain Hilbert space associated to a diagonal linear operator 93

3.6.5 Interpolation spaces associated to a diagonal linear operator . 94

3.7 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.7.1 Existence and uniqueness of the Bochner integral . . . . . . . 96

3.7.2 Definition of the Bochner integral . . . . . . . . . . . . . . . . 97

4 Semigroups of bounded linear operators 99

4.1 Definition of a semigroup of bounded linear operators . . . . . . . . . 99

4.2 Types of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 The generator of a semigroup . . . . . . . . . . . . . . . . . . . . . . 100

4.4 Global a priori bounds for semigroups . . . . . . . . . . . . . . . . . . 101

4.5 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . 102

4.5.1 A priori bounds for strongly continuous semigroups . . . . . . 102

4.5.1.1 The Baire category theorem on complete metric spaces102

4.5.1.2 The uniform boundedness principle . . . . . . . . . . 103

4.5.1.3 Local a priori bounds . . . . . . . . . . . . . . . . . 103

4.5.1.4 Global a priori bounds . . . . . . . . . . . . . . . . . 104

4.5.2 Existence of solutions of linear ordinary differential equationsin Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5.3 Pointwise convergence in the space of bounded linear operators 106

4.5.4 Domains of generators of strongly continuous semigroups . . . 107

4.5.5 Generators of strongly continuous semigroups . . . . . . . . . 108

4.5.6 A generalization of matrix exponentials to infinite dimensions 111

4.5.7 A characterization of strongly continuous semigroups . . . . . 111

4.6 Uniformly continuous semigroups . . . . . . . . . . . . . . . . . . . . 112

4.6.1 Matrix exponential in Banach spaces . . . . . . . . . . . . . . 113

4.6.2 Continuous invertibility of bounded linear operators in Banachspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.6.3 Generators of uniformly continuous semigroup . . . . . . . . . 116

4.6.4 A characterization result for uniformly continuous semigroups 117

4.6.5 An a priori bound for uniformly continuous semigroups . . . . 118

4.7 The Hille-Yosida theorem . . . . . . . . . . . . . . . . . . . . . . . . 118

4.8 Semigroups generated by diagonal operators . . . . . . . . . . . . . . 125

4.8.1 Smoothing effect of the semigroup . . . . . . . . . . . . . . . . 130

4.8.2 Semigroup generated by the Laplace operator . . . . . . . . . 132

8 CONTENTS

II Foundations in probability theory 135

5 Random variables with values in infinite dimensional spaces 1375.1 General measure and probability spaces . . . . . . . . . . . . . . . . . 137

5.1.1 Uniqueness theorem for measures . . . . . . . . . . . . . . . . 1375.1.2 Independence on probability spaces . . . . . . . . . . . . . . . 1405.1.3 Factorization lemma for conditional expectations . . . . . . . 141

5.2 Borel sigma-algebras on normed vector spaces . . . . . . . . . . . . . 1435.2.1 The Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . 1435.2.2 Norm representations in normed vector spaces . . . . . . . . . 1445.2.3 Linear characterization of the Borel sigma-algebra . . . . . . . 145

5.3 Probability measures on normed vector spaces . . . . . . . . . . . . . 1465.3.1 Fourier transform of a measure . . . . . . . . . . . . . . . . . 146

5.3.1.1 Characteristic functionals . . . . . . . . . . . . . . . 1465.3.1.2 Fourier transform on separable normed vector spaces 1475.3.1.3 Almost surely separably supported . . . . . . . . . . 1485.3.1.4 Trace set . . . . . . . . . . . . . . . . . . . . . . . . 1495.3.1.5 Fourier transform on normed vector spaces . . . . . . 152

5.3.2 Covariances on normed vector spaces . . . . . . . . . . . . . . 1555.3.2.1 Regularities for correlations on normed vector spaces 1555.3.2.2 Covariances on normed vector spaces . . . . . . . . . 158

5.4 Probability measures on Hilbert spaces . . . . . . . . . . . . . . . . . 1595.4.1 Nuclear operators on Hilbert spaces . . . . . . . . . . . . . . . 159

5.4.1.1 Rank-1 operators on inner product spaces . . . . . . 1605.4.1.2 Traces of nuclear operators . . . . . . . . . . . . . . 1615.4.1.3 Absolute value operators . . . . . . . . . . . . . . . . 162

5.4.2 Covariances on Hilbert spaces . . . . . . . . . . . . . . . . . . 1645.4.3 Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . 167

5.5 Gaussian measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.5.1 Gaussian measures on normed vector spaces . . . . . . . . . . 168

5.5.1.1 Fourier transform of a Gaussian measure . . . . . . . 1705.5.1.2 Covariance of a Gaussian measure . . . . . . . . . . 170

5.5.2 Gaussian measures on Hilbert spaces . . . . . . . . . . . . . . 1715.5.2.1 Karhunen-Loeve expansion . . . . . . . . . . . . . . 1715.5.2.2 Fourier transform of a Gaussian measure . . . . . . . 1725.5.2.3 Construction of Gaussian measures on Hilbert spaces 1725.5.2.4 Class exercise on Gaussian distributed random variables1765.5.2.5 Karhunen-Loeve expansion for Brownian motion . . 178

CONTENTS 9

6 Stochastic processes 1876.1 Hilbert space valued stochastic processes . . . . . . . . . . . . . . . . 187

6.1.1 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.1.2 Standard Wiener processes . . . . . . . . . . . . . . . . . . . . 1916.1.3 Pseudo inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.2 Space-time white noise and Brownian sheet . . . . . . . . . . . . . . . 1976.2.1 Derivative of a Brownian sheet . . . . . . . . . . . . . . . . . 197

6.3 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.3.1 Lenglart’s inequality . . . . . . . . . . . . . . . . . . . . . . . 1986.3.2 Modifications and indistinguishability . . . . . . . . . . . . . . 2026.3.3 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.3.4 Construction of the stochastic integral . . . . . . . . . . . . . 2056.3.5 On the density of elementary processes . . . . . . . . . . . . . 2146.3.6 Elementary processes revisited . . . . . . . . . . . . . . . . . . 2196.3.7 Cylindrical Wiener process . . . . . . . . . . . . . . . . . . . . 220

III Stochastic Partial Differential Equations (SPDEs) 223

7 Solutions of SPDEs 2257.1 Existence, uniqueness and properties of mild solutions of SPDEs . . . 225

7.1.1 Mild solutions of SPDEs . . . . . . . . . . . . . . . . . . . . . 2257.1.2 A setting for SPDEs with globally Lipschitz continuous non-

linearities* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.1.3 A strong perturbation estimate for SPDEs . . . . . . . . . . . 2287.1.4 Uniqueness of mild solutions of SPDEs . . . . . . . . . . . . . 233

7.1.4.1 Uniqueness of predictable mild solutions of SEEs withglobally Lipschitz continuous coefficients . . . . . . . 233

7.1.4.2 Uniqueness of left-continuous mild solutions of SEEswith semi-globally Lipschitz continuous coefficients . 233

7.1.5 Existence and regularity of mild solutions of SPDEs . . . . . . 2377.1.6 A priori bounds for mild solutions of SPDEs . . . . . . . . . . 237

7.1.6.1 A priori bounds . . . . . . . . . . . . . . . . . . . . . 2377.1.6.2 A priori bounds revisited . . . . . . . . . . . . . . . 2387.1.6.3 Strengthened regularity . . . . . . . . . . . . . . . . 240

7.1.7 Temporal-regularity of solution processes of SPDEs . . . . . . 2427.1.8 Existence of continuous solutions . . . . . . . . . . . . . . . . 242

7.2 Examples of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

10 CONTENTS

7.2.1 Second order SPDEs* . . . . . . . . . . . . . . . . . . . . . . 245

IV Numerical Analysis of SPDEs 251

8 Strong numerical approximations for SPDEs 253

8.1 Spatial spectral Galerkin approximations for SPDEs . . . . . . . . . . 253

8.1.1 Galerkin projections . . . . . . . . . . . . . . . . . . . . . . . 253

8.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8.1.3 A strong numerical approximation result for spectral Galerkinapproximations of SPDEs . . . . . . . . . . . . . . . . . . . . 260

8.2 Temporal numerical approximations for SPDEs . . . . . . . . . . . . 265

8.2.1 Euler type approximations for SPDEs . . . . . . . . . . . . . . 265

8.2.1.1 Exponential Euler method . . . . . . . . . . . . . . . 265

8.2.1.2 Accelerated exponential Euler method . . . . . . . . 267

8.2.1.3 Linear-implicit Euler method . . . . . . . . . . . . . 268

8.2.1.4 Linear-implicit Crank-Nicolson-Euler method . . . . 269

8.2.2 Nonlinearity-stopped Euler type approximations for SPDEs . . 270

8.2.2.1 Nonlinearity-stopped exponential Euler method . . . 271

8.2.2.2 Nonlinearity-stopped linear-implicit Euler method . . 272

8.2.3 Milstein type approximations for SPDEs . . . . . . . . . . . . 273

8.2.3.1 Exponential Milstein method . . . . . . . . . . . . . 273

8.2.3.2 Linear-implicit Milstein method . . . . . . . . . . . . 274

8.2.3.3 Linear-implicit Crank-Nicolson-Milstein method . . . 276

8.2.4 Strong convergence analysis for exponential Euler approxima-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

8.3 Noise approximations for SPDEs . . . . . . . . . . . . . . . . . . . . 288

8.3.1 Noise perturbation estimates . . . . . . . . . . . . . . . . . . . 288

8.3.2 Noise approximations for SPDEs . . . . . . . . . . . . . . . . 289

8.4 Full discretizations for SPDEs . . . . . . . . . . . . . . . . . . . . . . 292

8.4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.4.2 Full-discrete spectral Galerkin exponential Euler method forSPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.4.3 Full-discrete spectral Galerkin linear-implicit Euler method forSPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

8.4.4 Full-discrete spectral Galerkin nonlinearity-stopped exponen-tial Euler method for SPDEs . . . . . . . . . . . . . . . . . . . 297

CONTENTS 11

8.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicitEuler method for SPDEs . . . . . . . . . . . . . . . . . . . . . 298

9 Solutions to selected exercises 3039.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

9.1.1 Solution to Exercise 2.2.6 . . . . . . . . . . . . . . . . . . . . 303

12 CONTENTS

Part I

Foundations inmathematical analysis

13

Chapter 1

Gronwall-type inequalities

This chapter is based on Section 7.1 in Henry [5].

1.1 Properties of the beta and the gamma func-

tion

For completeness we first recall the definition of the gamma function and the betafunction.

Definition 1.1.1 (Gamma function*). We denote by Γ: p0,8q Ñ p0,8q the functionwith the property that for all x P p0,8q it holds that

Γpxq “

ż 8

0

tpx´1q e´t dt (1.1)

and we call Γ the gamma function.

Definition 1.1.2 (Beta function*). We denote by B : p0,8q2 Ñ p0,8q the functionwith the property that for all x, y P p0,8q it holds that

Bpx, yq “

ż 1

0

tpx´1qp1´ tqpy´1q dt (1.2)

and we call B the beta function.

15

16 CHAPTER 1. GRONWALL-TYPE INEQUALITIES

1.1.1 Functional equation of the gamma function

Lemma 1.1.3 (Basic properties of the gamma function and the Beta function*).For all x, y P p0,8q, n P N0 it holds that

Bpx, yq “ Bpy, xq “Γpxq ¨ Γpyq

Γpx` yq“

ż 8

0

tpx´1q

p1` tqpx`yqdt, (1.3)

Γpn` 1q “ n! and Γpx` 1q “ x ¨ Γpxq . (1.4)

Proof of Lemma 1.1.3*. First, observe that the integral transformation theorem en-sures that for all x, y P p0,8q it holds that

Bpx, yq “

ż 1

0

tpx´1qp1´ tqpy´1q dt “

ż 8

1

“

1t

‰px´1q “1´ 1

t

‰py´1q 1t2dt

“

ż 8

1

tp´x´1q“

t´1t

‰py´1qdt “

ż 8

1

tp´xýq pt´ 1qpy´1q dt

“

ż 8

0

pt` 1qp´xýq tpy´1qdt “

ż 8

0

tpy´1q

pt` 1qpx`yqdt.

(1.5)

Moreover, note that for all x P p0,8q it holds that

Γpx` 1q “

ż 8

0

tppx`1q´1q e´t dt “ ´

ż 8

0

tx“

é´t‰

dt

“ ´

ˆ

“

txe´t‰t“8

t“0´ x

ż 8

0

tpx´1q e´t dt

˙

“ x

ż 8

0

tpx´1q e´t dt “ x ¨ Γpxq.

(1.6)

The proof of Lemma 1.1.3 is thus completed.

1.1.2 Monotonicity properties of the gamma and the betafunction

Lemma 1.1.4 (Montonicity property of the gamma function*). It holds that

limxÑ8

Γ1pxq “ 8 (1.7)

and there exists a real number C P p0,8q such that for all x, y P rC,8q with x ď yit holds that Γpxq ď Γpyq.

1.1. PROPERTIES OF THE BETA AND THE GAMMA FUNCTION 17

Proof of Lemma 1.1.4*. Observe that for all x P p0,8q it holds that

Γ1pxq “

ż 8

0

lnptq tpx´1q e´t dt

“

ż 1

0

lnptq tpx´1q e´t dt`

ż e

1

lnptq tpx´1q e´t dt`

ż 8

e


ě inftPp0,1q

“

lnptq tpx´1q e´t‰

`

ż 8

e


ě inftPp0,1q

“

lnptq tpx´1q‰

`

ż 8

e

tpx´1q e´t dt.

(1.8)

This proves that

limxÑ8

Γ1pxq ě inftPp0,1q

rlnptq ts ` limxÑ8

ż 8

e

tpx´1q e´t dt “ 8. (1.9)


Lemma 1.1.5 (Monotonicity of the beta function*). For all x, y, x, y P p0,8q withx ď x and y ď y it holds that

Bpx, yq ď Bpx, yq. (1.10)

Proof of Lemma 1.1.5*. Note that for all θ P p0, 1q, x, x P R with x ď x it holdsthat

θx ď θx. (1.11)

Combining this with Definition 1.1.2 completes the proof of Lemma 1.1.5.

1.1.3 Estimates for terms containing the beta or the gammafunction

Lemma 1.1.6 (An upper bound for the beta function*). Let x, y P p0,8q withpx´ 1q py ´ 1q ě 0. Then

Bpx, yq “

ż 1

0

p1´ tqpx´1q tpy´1q dt ď

ż 1

0

tpx`y´2q dt “

#

8 : x` y ď 11

px`y´1q: x` y ą 1

. (1.12)


Proof of Lemma 1.1.6*. First, observe that the equalities in (1.12) are clear. It thusremains to prove the inequality in (1.12). For this we assume w.l.o.g. that x` y ą 1,that x ‰ 1 and that y ‰ 1 (otherwise also the inequality in (1.12) is clear). Theassumption that px´ 1q py ´ 1q ě 0 hence shows that px´ 1q py ´ 1q ą 0 and thatpy´1qpx´1q

P p0,8q. Combining this with Holder’s inequality proves that

ż 1

0

p1´ tqpx´1q tpy´1q dt

ď

„ż 1

0

p1´ tqpx´1qr1`py´1qpx´1qs dt

´

r1`py´1qpx´1qs

´1¯

„ż 1

0

tpy´1qr1`px´1qpy´1q s dt

´

r1`px´1qpy´1q s

´1¯

“

„ż 1

0

p1´ tqpx`y´2q dt

´

r1`py´1qpx´1qs

´1¯

„ż 1

0

tpx`y´2q dt

´

r1`px´1qpy´1q s

´1¯

“

ż 1

0

tpx`y´2q dt.

(1.13)


Remark 1.1.7. Lemma 1.1.6, in particular, shows that for all x, y P p0,8q withpx´ 1q py ´ 1q ě 0 it holds that

Bpx, yq ď

ż 1

0

tpx`y´2q dt. (1.14)

However, it is not true that for all x, y P p0,8q it holds that

Bpx, yq ď

ż 1

0

tpx`y´2q dt. (1.15)

Indeed, observe that

limxŒ0

Bpx, 2´ xq “ limxŒ0

ż 1

0

p1´ tqpx´1q tp2´xq dt ě limxŒ0

ż 1

12

p1´ tqpx´1q tp2´xq dt

ě

„

1

2

p2´xq

limxŒ0

ż 1

12

p1´ tqpx´1q dt “ 8 ą 1 “

ż 1

0

tp2´2q dt.

(1.16)

Exercise 1.1.8 (*). Prove that for all c P r0,8q, ε P p0,8q it holds that

8ÿ

n“1

cn

Γpnεqă 8. (1.17)

1.1. PROPERTIES OF THE BETA AND THE GAMMA FUNCTION 19

Exercise 1.1.9 (*). Prove that for all c P r0,8q, ε P p0,8q it holds that

8ÿ

n“1

cn„

nś

k“1

B`

kε, ε˘

ă 8. (1.18)

Exercise 1.1.10 (*). Prove that for all c P r0,8q, ε, δ, ρ P p0,8q it holds that

8ÿ

n“1

cn„

n´1ś

k“0

B`

ε` kδ, ρ˘

ă 8. (1.19)

Proposition 1.1.11 (Bounds for the gamma function*). It holds for all n P N that

”n

3

ın

ď

”n

e

ın

ă e”n

e

ın

ď n! ď e

„

n` 1

e

n`1

(1.20)

and it holds for all n P N0 that

nn ď en ¨ n! ď pn` 1qpn`1q (1.21)

Proof of Proposition 1.1.11*. Observe that for all n P N0 it holds that

lnpn!q “ ln`

n ¨ pn´ 1q ¨ . . . ¨ 2 ¨ 1˘

“

nÿ

k“1

lnpkq “nÿ

k“1

ż k`1

k

lnpkq dx

ď

nÿ

k“1

ż k`1

k

lnpxq dx “

ż n`1

1

lnpxq dx “ rx lnpxq ´ xsx“n`1x“1

“ pn` 1q lnpn` 1q ´ pn` 1q ` 1 “ pn` 1q lnpn` 1q ´ n.

(1.22)

Moreover, (1.22) ensures for all n P N that

lnpn!q “nÿ

k“1

lnpkq “nÿ

k“2

ż k

k´1

lnpkq dx ěnÿ

k“2

ż k

k´1

lnpxq dx “

ż n

1

lnpxq dx

“ rx lnpxq ´ xsx“nx“1 “ n lnpnq ´ n` 1.

(1.23)

Combining this and (1.22) proves for all n P N that

nn

en´1“ en lnpnqń`1

ď n! ď epn`1q lnpn`1qń“pn` 1qpn`1q

en. (1.24)

The proof of Proposition 1.1.11 is thus completed.


1.2 Integral operators related to the beta function

Lemma 1.2.1 (A scaling property of the beta function*). For all β, γ P p0,8q,r, t P r0,8q with r ď t it holds that

ż t

r

pt´ sqpβ´1qps´ rqpγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq. (1.25)

Proof of Lemma 1.2.1*. Note that for all β, γ P p0,8q, r, t P r0,8q with r ď t itholds that

ż t

r

pt´ sqpβ´1qps´ rqpγ´1q ds “

ż pt´rq

0

pt´ r ´ sqpβ´1q spγ´1q ds

“ pt´ rqpβ`γ´1q

ż 1

0

p1´ sqpβ´1q spγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq.

(1.26)


The next estimate, inequality (1.27) in Lemma 1.2.2, is an immediate consequenceof Lemma 1.2.1.

Lemma 1.2.2. Let α, γ, τ P R, T P rτ,8q, u PMpBprτ, T sq,Bpr0,8sqq, β P p0,8qsatisfy α ` γ ą 1. Then it holds for all t P rτ, T s that

ż t

τ

pt´ sqpβ´1qps´ τqpγ´1q upsq ds

ď pt´ τqpα`β`γ´2qB`

β, α ` γ ´ 1˘

«

supsPpτ,tq

upsq

ps´ τqpα´1q

ff

.

(1.27)

We need a further estimate for the integral operator appearing on the left handside of (1.27). This is the subject of the next lemma.

1.2. INTEGRAL OPERATORS RELATED TO THE BETA FUNCTION 21

Lemma 1.2.3 (Iterations of an integral operator). Let α, γ, τ P R, T P rτ,8q, b Pr0,8q, β P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq Ñ MpBprτ, T sq,Bpr0,8sqq, assumethat mintα, βu`γ ą 1, and assume that for all u PMpBprτ, T sq,Bpr0,8sqq, t P rτ, T sit holds that

`

Bpuq˘

ptq “ b

ż t

τ

pt´ sqpβ´1qps´ τqpγ´1q upsq ds. (1.28)

Then it holds for all n P N, t P rτ, T s, u PMpBprτ, T sq,Bpr0,8sqq that

`

Bnpuq

˘

ptq

ď bn pt´ τqpα´1`npβ`γ´1qq

«

n´1ź

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ff«

supsPpτ,tq

upsq

ps´τqpα´1q

ff

(1.29)

and that

`

Bnpuq

˘

ptq ď bn pt´ τqpn´1qrγ´1s`

«

n´1ź

k“1

B`

β, kpβ ´ r1´ γs`q˘

ff

¨t

∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q

ps´ τqpγ´1q upsq ds.

(1.30)

Proof of Lemma 1.2.3. Estimate (1.29) is an immediate consequence of Lemma 1.2.2.It thus remains to prove estimate (1.30). For this we assume in the following w.l.o.g.that τ “ 0. Then note that Lemma 1.2.1 implies that for all u PMpBpr0, T sq,Bpr0,8sq,t P r0, T s, b P r0,8q, β, γ P p0,8q with β ` γ ą 1 it holds that

b

ż t

0

pt´ sqpβ´1q spγ´1q

„

b

ż s

0

ps´ rqpβ´1q rpγ´1q uprq dr

ds

“ b b

ż t

0

ż s

0

pt´ sqpβ´1q spγ´1qps´ rqpβ´1q rpγ´1q uprq dr ds

“ b b

ż t

0

rpγ´1q uprq

„ż t

r

pt´ sqpβ´1q spγ´1qps´ rqpβ´1q ds

dr

ď b b trγ´1s`ż t

0

rpγ´1q uprq

„ż t

r

pt´ sqpβ´1qps´ rqpβ´1`mintγ´1,0uq ds

dr

ď B`

β, β ´ r1´ γs`˘

b b trγ´1s`ż t

0

pt´ rqpβ`β´r1´γs`´1q rpγ´1q uprq dr.

(1.31)


Iterating (1.31) shows that for all u PMpBpr0, T sq,Bpr0,8sq, t P r0, T s, n P t2, 3, . . . uit holds that

`

Bnpuq

˘

ptq ď bn tpn´1qrγ´1s`t

∫0pt´ sqpβ`pn´1qpβ´r1´γs`q´1q spγ´1q upsq ds

¨

«

n´1ź

k“1

B`


ff

.

(1.32)


1.3 Generalized exponential-type functions

Definition 1.3.1 (Generalized exponential-type functions*). We denote by Er : r0,8q Ñr0,8q, r P p0,8q, Er : r0,8q Ñ r0,8q, r P p0,8q, and Er : r0,8q Ñ r0,8q,r P p0,8q, the functions with the property that for all r P p0,8q, x P r0,8q itholds that

Errxs “8ÿ

n“0

xnr

Γpnr ` 1q, Errxs “ Er

”

pxΓprqq1r

ı

“

8ÿ

n“0

pxΓprqqn

Γpnr ` 1q(1.33)

and Errxs “b

Errx2s “

«

8ÿ

n“0

px2Γprqqn

Γpnr ` 1q

ff12

. (1.34)

Lemma 1.3.2. Let r P p0,8q, x P r0,8q. Then

8ÿ

n“0

xnr

Γpnr ` 1qďemaxtx,1u maxtx, 1u r1rs1

infsPp0,8q Γpsq. (1.35)

Proof of Lemma 1.3.2. First, observe that

8ÿ

n“0

xnr

Γpnr ` 1q“

8ÿ

k“0

ÿ

nPN0,kďnrăk`1

xnr

Γpnr ` 1q

“ÿ

nPN0,nră1

xnr

Γpnr ` 1q`

8ÿ

k“1

ÿ

nPN0,kďnrăk`1

xnr

Γpnr ` 1q

ďÿ

nPN0,nră1

xnr

infsPp0,8q Γpsq`

8ÿ

k“1

ÿ

nPN0,kďnrăk`1

rmaxtx, 1uspk`1q

Γpk ` 1q.

(1.36)

1.4. GENERALIZED TIME-CONTINUOUS GRONWALL-TYPE INEQUALITIES23

This shows that

8ÿ

n“0

xnr

Γpnr ` 1qď

1

infsPp0,8q Γpsq

»

—

–

8ÿ

k“0

ÿ

nPN0,kďnrăk`1

rmaxtx, 1uspk`1q

Γpk ` 1q

fi

ffi

fl

“maxtx, 1u

infsPp0,8q Γpsq

«

8ÿ

k“0

rmaxtx, 1usk

Γpk ` 1q#N0ptn P N0 : k ď nr ă k ` 1uq

ff

ď#N0ptn P N0 : nr ă 1uqmaxtx, 1u

infsPp0,8q Γpsq

«

8ÿ

k“0

rmaxtx, 1usk

k!

ff

“emaxtx,1u#N0ptn P N0 : n ă 1ruqmaxtx, 1u

infsPp0,8q Γpsq“

maxtx, 1u emaxtx,1u#N0pr0, 1rqq

infsPp0,8q Γpsq.

(1.37)


1.4 Generalized time-continuous Gronwall-type in-

equalities

Lemma 1.4.1 (Main idea in the proof of the generalized Gronwall inequality). Letτ P R, T P rτ,8q, b P MpBprτ, T s2q,Bpr0,8sqq, a, e P MpBprτ, T sq,Bpr0,8sqq,B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy that for all t P rτ, T s,u PMpBprτ, T sq,Bpr0,8sqq it holds that

`

Bpuq˘

ptq “t

∫τbpt, squpsq ds, (1.38)

and assume that e ď a`Bpeq. Then it holds for all n P N that

e ďn´1ÿ

k“0

Bkpaq `Bn

peq. (1.39)

Proof of Lemma 1.4.1. Estimate (1.39) follows immediately from an iterated applica-tion of the assumption e ď a`Bpeq and from the fact that B is monotone in the sensethat for all u, u P MpBprτ, T sq,Bpr0,8sqq with u ď u it holds that Bpuq ď Bpuq.The proof of Lemma 1.4.1 is thus completed.


Next we present the generalized Gronwall inequalities. They are modified versionsof the estimates in Section 7.1 in Henry [5].

Theorem 1.4.2. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β, γ P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy β ` γ ą

1 andşT

τps´ τqpγ´1q epsq ds ă 8, assume that for all u P MpBprτ, T sq,Bpr0,8sqq,

t P rτ, T s it holds that

pBuqptq “ bt

∫τpt´ sqpβ´1q

ps´ τqpγ´1q upsq ds, (1.40)

and assume that for all t P rτ, T s it holds that

eptq ď aptq ` bt

∫τpt´ sqpβ´1q

ps´ τqpγ´1q epsq ds. (1.41)

Then it holds for all t P rτ, T s that

eptq ď8ÿ

n“0

`

Bnpaq

˘

ptq ď aptq `8ÿ

n“1

bn pt´ τqpn´1qrγ´1s`„

n´1ś

k“1

B`


¨t

∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q

ps´ τqpγ´1q apsq ds (1.42)

and it holds for all t P pτ, T s, α P p0,8q with α ` γ ą 1 that

eptq ď8ÿ

n“0

`

Bnpaq

˘

ptq ď aptq (1.43)

`

«

supsPpτ,tq

apsq

ps´τqpα´1q

ff

8ÿ

n“1

bn pt´ τqpα´1`npβ`γ´1qq

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

looooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooon

ă8

.

Proof of Theorem 1.4.2. W.l.o.g. we assume that τ “ 0. Lemma 1.4.1 implies thatfor all n P N0 it holds that

e ď a`Bpaq `B2paq ` . . .`Bn

paq `Bpn`1qpeq “

«

nÿ

k“0

Bkpaq

ff

`Bpn`1qpeq. (1.44)

Next we note that inequality (1.30) in Lemma 1.2.3 together with the assumptionthat @ t P r0, T s :

şt

0spγ´1q epsq ds ă 8 and the fact that

@ c P r0,8q, r P p0,8q : limnÑ8

„

cn

Γppn´ 1qrq

“ 0 (1.45)


(see Exercise 1.1.8) implies that for all t P p0, T s it holds that

limnÑ8

`

Bnpeq

˘

ptq

ď limnÑ8

«

bn tβ´1`pn´1qpβ`γ´1qt

∫0spγ´1qepsq ds

«

n´1ź

k“1

B`


ffff

ď∫ t0 spγ´1q epsq ds

tγlimnÑ8

«

“

tpβ`γ´1q b‰n

«

n´1ź

k“1

B`


ffff


tγlimnÑ8

«

“

maxp1,Γpβqq tpβ`γ´1q b‰n

«

n´1ź

k“1

Γpkpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

ffff


tγ

¨ limnÑ8

«

“

|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n

Γ`

β ` pn´ 1qpβ ´ r1´ γs`q˘

„

n´2ś

k“1

Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

ff


tγ

„

8ś

k“1

Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

¨ limnÑ8

«

“

|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n

Γ`

pn´ 1qpβ ´ r1´ γs`q˘

ff

“ 0.

(1.46)

This and (1.44) prove the first inequalities in (1.42) and (1.43). Estimate (1.30) inLemma 1.2.3 proves the second inequality in (1.42). Furthermore, estimate (1.29)in Lemma 1.2.3 proves the second inequality in (1.43). Finally, observe that Exer-cise 1.1.10 implies that for all t P pτ, T s it holds that

8ÿ

n“1

bn pt´ τqnpβ`γ´1q

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

“

8ÿ

n“1

”

b pt´ τqpβ`γ´1qın

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ă 8.

(1.47)

The proof of Theorem 1.4.2 is thus completed.

1.4.1 Gronwall-type inequalities with singularities at initialtime

The next result, Corollary 1.4.3, specialises estimate (1.42) in Theorem 1.4.2 to thecase where γ “ 1; see Lemma 7.1.1 in Henry [5].


Corollary 1.4.3. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β P p0,8q satisfy

şT

τepsq ds ă 8 and assume that for all t P rτ, T s it holds that

eptq ď aptq ` bt

∫τpt´ sqpβ´1q epsq ds. (1.48)


eptq ď aptq ` rΓpβq bs1β

ż t

τ

E1β“

pt´ sq rΓpβq bs1β‰

apsq ds. (1.49)

Proof of Corollary 1.4.3. Inequality (1.42) in Theorem 1.4.2 with γ “ 1 shows thatfor all t P rτ, T s it holds that

eptq ď aptq `

ż t

τ

«

8ÿ

n“1

rΓpβq bsn pt´ sqpnβ´1q

Γpnβq

ff

apsq ds

“ aptq ` rΓpβq bs1β

ż t

τ

«

8ÿ

n“1

“

pt´ sq rΓpβq bs1β‰pnβ´1q

Γpnβq

ff

apsq ds.

(1.50)

Next note that Lemma 1.1.3 shows that for all x P p0,8q it holds that

E1βpxq “8ÿ

n“1

nβxpnβ´1q

Γpnβ ` 1q“

8ÿ

n“1

xpnβ´1q

Γpnβq. (1.51)

Combining this with (1.50) completes the proof of Corollary 1.4.3.

The next result, Corollary 1.4.4, specialises estimate (1.43) in Theorem 1.4.2 tothe case where the function a in Theorem 1.4.2 satisfies aptq “ c tpα´1q for all t P pτ, T sand some c P r0,8q; see Exercise 3 in Henry [5]. Corollary 1.4.4 follows immediatelyfrom (1.43) in Theorem 1.4.2 and from Exercise 1.1.10.

Corollary 1.4.4. Let τ P R, a, b P r0,8q, T P rτ,8q, α, β, γ P p0,8q, a, e P

MpBprτ, T sq,Bpr0,8sqq satisfy mintα, βu ` γ ą 1 andşT

τps´ τqpγ´1q epsq ds ă 8

and assume that for all t P pτ, T s it holds that

eptq ď a pt´ τqpα´1q` b

t

∫τpt´ sqpβ´1q

ps´ τqpγ´1q epsq ds. (1.52)

Then it holds for all t P pτ, T s that

eptq ď a pt´ τqpα´1q8ÿ

n“0

bn pt´ τqnpβ`γ´1q

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ă 8.

(1.53)


The next result, Corollary 1.4.5, specialises Corollary 1.4.4 to the case γ “ 1; cf.Exercise 4 in Henry [5]. Corollary 1.4.5 follows immediately from Corollary 1.4.4.

Corollary 1.4.5. Let τ P R, T P rτ,8q, e PMpBprτ, T sq,Bpr0,8sqq, a, b P r0,8q,α, β P p0,8q satisfy

şT

τepsq ds ă 8 and assume that for all t P pτ, T s it holds that

eptq ď a pt´ τqpα´1q` b

t


Then it holds for all t P pτ, T s that

eptq ď a pt´ τqpα´1q8ÿ

n“0

bn pt´ τqnβ„

n´1ś

k“0

B`

β, α ` kβ˘

ă 8. (1.55)

1.4.2 Gronwall-type inequalities without singularities at ini-tial time

In the remainder of these lecture notes we will often use the following Gronwall-typeestimate; see Lemma 7.1.1 in Henry [5].

Corollary 1.4.6 (*). Let τ P R, T P rτ,8q, e PMpBprτ, T sq,Bpr0,8sqq, β P p0,8q,a, b P r0,8q satisfy

şT

τepsq ds ă 8 and assume for all t P rτ, T s that

eptq ď a` bt



eptq ď a ¨ Eβ

”

pt´ τq pbΓpβqq1βı

“ a ¨ Eβ

“

pt´ τqβ b‰

. (1.57)

Proof of Corollary 1.4.6. Corollary 1.4.3 implies that for all t P rτ, T s it holds that

eptq ď a` rΓpβq bs1β

ż t

τ

E1β“

pt´ sq rΓpβq bs1β‰

a ds. (1.58)

The fundamental theorem of calculus hence shows that for all t P rτ, T s it holds that

eptq ď a

„

1` limεŒ0

ż t

τ`ε

rΓpβq bs1β E1β

“

ps´ τq rΓpβq bs1β‰

ds

“ a

„

1` limεŒ0

“

Eβ

“

pt´ τq rΓpβq bs1β‰

´ Eβ

“

ε rΓpβq bs1β‰‰

“ a“

1``

Eβ

“


´ Eβ

“

0‰˘‰

“ aEβ

“


.

(1.59)

The proof of Corollary 1.4.6 is thus completed.


Chapter 2

Nonlinear functions and nonlinearspaces

Most of this chapter is based on Da Prato & Zabczyk [3] and Prevot & Rockner [18].

2.1 Sets and relations

Definition 2.1.1 (Power set*). Let A be a set. Then we denote by PpAq the powerset of A (the set of all subsets of A).

Definition 2.1.2 (Sets of numbers*). We denote by

N “ t1, 2, 3, . . . u (2.1)

the set of natural numbers, we denote by

N0 “ NY t0u “ t0, 1, 2, . . . u (2.2)

the union of t0u and the set of natural numbers, we denote by

Z “ t0, 1,´1, 2,´2, . . . u (2.3)

the set of integer numbers, we denote by Q the set of rational numbers, we denoteby R the set of real numbers and we denote by C the set of complex numbers.

Note that Definition 2.1.2, in particular, ensures that N Ă N0 Ă Z Ă Q Ă R Ă

C. For completeness we recall the notion of a relation in the following definition,

29

30 CHAPTER 2. NONLINEAR FUNCTIONS AND NONLINEAR SPACES

Definition 2.1.3. Definition 2.1.3 is used in Subsection 2.1.1 below to recall the notionof a function.

Definition 2.1.3 (Relation*). Let A, B, and C be sets with the property that C ĎpAˆBq. Then and only then we say that pA,B,Cq is a relation (we say that pA,B,Cqis a relation on pA,Bq).

Definition 2.1.4 (*). Let „ “ pA,B,Cq be a relation, let a P A, and let b P B.Then we write a „ b if and only if pa, bq P C.

2.1.1 General functions

Definition 2.1.5 (Function*). Let pA,B,Cq be a relation with the property that forevery a P A there exists exactly one b P B such that pa, bq P C. Then and only thenwe say that pA,B,Cq is a function (we say that pA,B,Cq is a function from A toB).

Definition 2.1.6 (Set of functions*). Let A and B be sets. Then we denote byMpA,Bq the set of all functions from A to B.

Definition 2.1.7 (Domain of definition, codomain, image of a function*). Let Aand B be sets and let f PMpA,Bq. Then

(i) we denote by Dpfq the set given by Dpfq “ A and we call Dpfq the domain ofdefinition of f ,

(ii) we denote by codomainpfq the set given by codomainpfq “ B and we callcodomainpfq the codomain of f ,

(iii) we denote by impfq the set given by impfq “ fpAq “ tfpxq P B : x P Au andwe call impfq the image of f , and

(iv) we denote by graphpfq the set given by graphpfq “ tpx, yq P AˆB : y “ fpxquand we call graphpfq the graph of f .

2.1.2 Preordered sets

Definition 2.1.8 (Preordered set*). Let X be a set and let ĺ be a relation on pX,Xqwith the property that

(i) @x P X : x ĺ x (Reflexivity) and

(ii) @x, y, z P X :`

px ĺ y and y ĺ zq ñ px ĺ zq˘

(Transitivity).

Then and only then we say that pX,ĺq is a preordered set.

2.2. MEASURABLE FUNCTIONS 31

Definition 2.1.9 (Partially ordered set*). Let pX,ĺq be a preorder with the propertythat for all A,B P X with A ĺ B and B ĺ A it holds that A “ B (Antisymmetry).Then and only then we say that pX,ĺq is a partially ordered set.

Definition 2.1.10 (Directed set*). Let pX,ĺq be a preorder with the property that

@x, y P X : D z P X : px ĺ z and y ĺ zq . (2.4)

Then and only then we say that pX,ĺq is a directed set.

Definition 2.1.11 (Nets*). Let pX,ĺq be a directed set, let pE, Eq be a topologicalspace, and let φ : X Ñ E be a function from X to E. Then and only then we saythat φ is a net (we say that φ is a net from pX,ĺq to pE, Eq).

Definition 2.1.12 (Convergence of a net*). Let pX,ĺq be a directed set, let pE, Eqbe a topological space, let e P E, and let φ : X Ñ E be a net. Then we write

limxPpX,ĺq

φpxq “ e (2.5)

and we say that φ converges to e (we say that φ converges to e with respect to pX,ĺq)if and only if for every neighbourhood U Ď E of e there exists a y P X such that forall x P X with y ĺ x it holds that φpxq P U .

2.2 Measurable functions

We first recall the notion of a measurable mapping.

Definition 2.2.1 (Measurable functions*). Let pΩ1,F1q and pΩ2,F2q be measurablespaces and let f : Ω1 Ñ Ω2 be a function with the property that for all F P F2 it holdsthat

f´1pF q P F1. (2.6)

Then and only then we say that f is F1/F2-measurable.

Definition 2.2.2 (Set of all measurable functions*). Let pΩ1,F1q and pΩ2,F2q bemeasurable spaces. Then we denote by MpF1,F2q the set of all F1/F2-measurablefunctions.

Definition 2.2.3 (Borel sigma-algebra*). Let pE, Eq be a topological space. Thenwe denote by BpEq the set given by BpEq “ σEpEq and we call BpEq the Borelsigma-algebra on pE, Eq.


2.2.1 Nonlinear characterization of the Borel sigma-algebra

Proposition 2.2.5 below demonstrates that if pE, dq is a metric space, then BpEq is thesmallest sigma-algebra with the property that every continuous real-valued functionis Borel measurable. We refer to the statement of Proposition 2.2.5 as nonlinearcharacterization of the Borel sigma-algebra. In the proof of Proposition 2.2.5 we usethe following notation.

Definition 2.2.4 (Distance of sets*). Let pE, dq be a metric space. Then we denoteby distd : PpEq ˆ PpEq Ñ r0,8s the function with the property that for all A,B P

PpEq it holds that

distdpA,Bq “

#

infaPA infbPB dpa, bq : A ‰ H and B ‰ H

8 : else. (2.7)

We now present the promised nonlinear characterization result for Borel sigma-algebras for metric spaces.

Proposition 2.2.5 (Nonlinear characterization of the Borel sigma-algebra*). LetpE, dq be a metric space. Then

BpEq “ σE`

pϕqϕPCpE,Rq˘

“ σEpϕ : ϕ P CpE,Rqq

“ σE`

ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq

(˘

.(2.8)

Proof of Proposition 2.2.5*. First of all, observe that for every ϕ P CpE,Rq it holdsthat ϕ is BpEq/BpRq-measurable. Hence, we obtain that

BpEq Ě σE`


(˘

. (2.9)

It thus remains to prove that

BpEq Ď σE`


(˘

. (2.10)

For this observe by that definition it holds that

BpEq “ σEptA P PpEq : A is an open set in pE, dquq . (2.11)


tA P PpEq : A is an open set in pE, dqu

Ď σE`


(˘

.(2.12)

2.2. MEASURABLE FUNCTIONS 33

For this let B Ă E be an open set in pE, dq and let ψ : E Ñ R be the function withthe property that for all x P E it holds that

ψpxq “ distdptxu, EzBq. (2.13)

Observe that ψ P CpE,Rq. This implies that

B “ ψ´1pp0,8qq Ď σE

`


(˘

. (2.14)


Exercise 2.2.6 (*). Let pΩ,Fq be a measurable space, let pE, dq be a metric space,and let f : Ω Ñ E be a function. Prove that f is F/BpEq-measurable if and only ifit holds for all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.

2.2.2 Pointwise limits of measurable functions

Lemma 2.2.7 (*). Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function,and let Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings such thatfor all ω P Ω it holds that Y pωq “ supnPNXnpωq. Then Y is F/BpRq-measurable.

Proof of Lemma 2.2.7*. Note that for all c P R it holds that

tY ď cu “

"

supnPN

Xn ď c

*

“č

nPN

tXn ď culoooomoooon

PF

P F . (2.15)


Lemma 2.2.8 (*). Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function,and let Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings suchthat for all ω P Ω it holds that lim supnÑ8 |Xnpωq ´ Y pωq| “ 0. Then Y is F/BpRq-measurable.

Proof of Lemma 2.2.8*. Note that Lemma 2.2.7 implies that for all c P R it holdsthat

tY ě cu “!

limnÑ8

Xn ě c)

“

"

lim supnÑ8

Xn ě c

*

“

#

limnÑ8

«

supmPtn,n`1,... u

Xm

ff

ě c

+

“č

nPN

#«

supmPtn,n`1,... u

Xm

ff

ě c

+

looooooooooooooomooooooooooooooon

PF

P F . (2.16)



The next corollary is an immediate consequence of Exercise 2.2.6 and Lemma 2.2.8;see, e.g., Proposition E.1 in [1] and Proposition A.1.3 in Prevot & Rockner [18].

Corollary 2.2.9 (*). Let pΩ,Fq be a measurable space, let pE, dq be a metric space,and let f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if thereexists a sequence gn : Ω Ñ E, n P N, of F/BpEq-measurable functions such that forall ω P Ω it holds that

lim supnÑ8

dpfpωq, gnpωqq “ 0. (2.17)

2.3 Strongly measurable functions

2.3.1 Simple functions

The idea of the Lebesgue integral for real valued functions is to approximate thefunction by suitable simpler functions and then to define the Lebesgue integral ofthe “complicated” function as the limit of the integrals of the simpler functions. Toperform this procedure we use the following definition.

Definition 2.3.1 (Simple functions*). Let pΩ1,F1q and pΩ2,F2q be measurable spacesand let f : Ω1 Ñ Ω2 be an F1/F2-measurable function with the property that the setfpΩ1q is finite. Then and only then we say that f is F1/F2-simple.

2.3.2 Separability

(Unfortunately) Measurable functions can, in general, not be approximated pointwise(see (2.17) in Corollary 2.2.9) by simple functions; see Theorem 2.3.10 below fordetails. To overcome this difficulty, we introduce the notion of a strongly measurablefunction; see Definition 2.3.6 below. In this notion the following definition is used.

Definition 2.3.2 (Separability*). Let pE, Eq be a topological space with the propertythat there exist an at most countable set A Ď E which satisfies A “ E. Then andonly then we say that E is separable (we say that pE, Eq is separable).

A topological space that is not separable is in a certain sense extremely large.This, in turn, can cause several serious difficulties in the analysis of such spaces. Anexample of a non-separable topological space can be found below. The next lemmaprovides an example for a separable topological space.

Lemma 2.3.3 (*). Let a, b P R with a ă b. Then pCpra, bs,Rq, ¨Cpra,bs,Rqq isseparable.

2.3. STRONGLY MEASURABLE FUNCTIONS 35

Proof of Lemma 2.3.3*. Observe that the set

"

f P Cpra, bs,Rq :

ˆ

Dn P N0 : Dλ0, . . . , λn P Q : @x P ra, bs : fpxq “nř

k“0

λkxk

˙*

(2.18)is a countable dense subset of Cpra, bs,Rq. The proof of Lemma 2.3.3 is thus com-pleted.

In the next lemma we provide a further simple example for a separable topologicalspace.

Lemma 2.3.4 (Trivial topology*). Let X be a set. Then it holds that the pairpX, tX,Huq is a separable topological space.

Proof of Lemma 2.3.4*. Throughout this proof assume w.l.o.g. that X ‰ H and letx P X. Next observe that txu is a finite dense subset of X. The proof of Lemma 2.3.4is thus completed.

Subspaces of separable metric spaces are separable too. This is the subject of thenext lemma.

Lemma 2.3.5 (*). Let pE, dq be a separable metric space and let A Ď E. Then themetric space pA, d|AÂq is separable.

Proof of Lemma 2.3.5*. W.l.o.g. we assume that A ‰ H. Let penqnPN Ď E be asequence of elements in E such that the set ten P E : n P Nu is dense in E. In thenext step let pfnqnPN Ď A be a sequence of elements in A such that for all n P N itholds that

dpfn, enq ď

#

0 : en P A

distdpA, tenuq `1

2n: en R A

. (2.19)


Next observe that for all v P Aztem P E : m P Nu, n P N it holds that

distdptvu, tf1, f2, . . . uq ď distdptvu, tfn, fn`1, . . . uq

“ infmPtn,n`1,... u

dpv, fmq

ď infmPtn,n`1,... u

rdpv, emq ` dpem, fmqs


„

dpv, emq ` distdpA, temuq `1

2m


„

2 dpv, emq `1

2m

ď 2

„

infmPtn,n`1,... u

dpv, emq

`1

2n

“ 2 distdptvu, ten, en`1, . . . uq `1

2n“

1

2n.

(2.20)

Combining this with the fact that @ v P AXtem : m P Nu : distdptvu, tf1, f2, . . . uq “ 0ensures that the set tfn P A : n P Nu is dense in A. The proof of Lemma 2.3.5 isthus completed.

2.3.3 Strongly measurable functions

Definition 2.3.6 (Strongly measurable functions*). Let pΩ,Fq be a measurablespace, let pE, dq be a metric space, and let f : Ω Ñ E be an F/BpEq-measurablefunction with the property that pfpΩq, d|fpΩqˆfpΩqq is separable. Then and only thenwe say that f is strongly F/pE, dq-measurable (we say that f is strongly measurable).

Let pΩ,Fq be a measurable space and let pE, dq be a separable metric space.Lemma 2.3.5 then shows that for every F/BpEq-measurable mapping f : Ω Ñ E itholds that f is also strongly F/pE, dq-measurable.

Exercise 2.3.7 (*). Give an example of a measurable space pΩ,Fq, of a metricspace pE, dEq, and of an F/BpEq-measurable function f : Ω Ñ E which is notstrongly F/pE, dEq-measurable. Prove that f is F/BpEq-measurable but not stronglyF/pE, dEq-measurable.


2.3.4 Pointwise approximations of strongly measurable func-tions

As mentioned above, measurable functions can, in general, not be approximatedpointwise by simple functions. However, strongly measurable functions can be ap-proximated pointwise by simple functions. This is the subject of the Theorem 2.3.10below (cf., e.g., Lemma 1.1 in Da Prato & Zabczyk [3] and Lemma A.1.4 in Prevot& Rockner [18]). In the proof of Theorem 2.3.10 the following two lemmas are used.

Lemma 2.3.8 (Projections in metric spaces*). Let pE, dq be a metric space, letn P N, e1, . . . , en P E, and let Ppe1,...,enq : E Ñ E be the function with the propertythat for all x P E it holds that

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dpek,xq“distdptxu,te1,...,enuqu. (2.21)

Then

(i) it holds that Ppe1,...,enq is BpEq/PpEq-measurable and

(ii) it holds for all x P E that

dpx, Ppe1,...,enqpxqq “ distdptxu, te1, . . . , enuq. (2.22)

Proof of Lemma 2.3.8*. Identity (2.22) is an immediate consequence of (2.21). LetD “ pD1, . . . , Dnq : E Ñ Rn be the function with the property that for all x P E itholds that

Dpxq “ pD1pxq, . . . , Dnpxqq “ pdpx, e1q, . . . , dpx, enqq . (2.23)

Observe that D is continuous and hence that D is BpEq/BpRnq-measurable. This


implies that for all k P t1, 2, . . . , nu it holds that

P´1pe1,...,enq

ptekuq “

x P E : Ppe1,...,enqpxq “ ek(

“

!

x P E : k “ min

l P t1, 2, . . . , nu : dpel, xq “ distdptxu, te1, . . . , enuq(

)

“

"

x P E : k “ min

"

l P t1, 2, . . . , nu : Dlpxq “ minuPt1,...,nu

Dupxq

**

“

"

x P E :

ˆ

Dkpxq ď minuPt1,...,nu

Dupxq and Dkpxq ă minuPt1,...,k´1u

Dupxq

˙*

“

"

x P E :

ˆ

@ l P t1, . . . , k ´ 1u : Dkpxq ă Dlpxq and@ l P t1, . . . , nu : Dkpxq ď Dlpxq

˙*

“

»

—

–

k´1č

l“1

tx P E : Dkpxq ă Dlpxquloooooooooooooomoooooooooooooon

PBpEq

fi

ffi

fl

č

»

—

–

nč

l“1

tx P E : Dkpxq ď Dlpxquloooooooooooooomoooooooooooooon

PBpEq

fi

ffi

fl

P BpEq.

(2.24)

This, in turn, implies that for all A P PpEq it holds that

P´1pe1,...,enq

pAq “ P´1pe1,...,enq

`

AX te1, . . . , enu˘

“ YfPAXte1,...,enu P´1pe1,...,enq

ptfuqlooooooomooooooon

PBpEq

P BpEq. (2.25)


Lemma 2.3.9 (*). Let pΩ,Fq be a measurable space, let pE, dq be a metric space,let f : Ω Ñ E be a function, and let gn : Ω Ñ E, n P N, be a sequence of stronglyF/pE, dq-measurable functions such that for all ω P Ω it holds that

lim supnÑ8


Then f is strongly F/pE, dq-measurable.

Proof of Lemma 2.3.9*. Corollary 2.2.9 ensures that f is F/BpEq-measurable. Itthus remains to prove that fpΩq is separable. This follows from Lemma 2.3.5 andfrom the fact that YnPNgnpΩq is separable. The proof of Lemma 2.3.9 is thus com-pleted.

We now present the promised pointwise approximation result for strongly mea-surable functions.


Theorem 2.3.10 (Approximations of strongly measurable functions*). Let pΩ,Fqbe a measurable space, let pE, dq be a metric space, and let f : Ω Ñ E be a function.Then the following four statements are equivalent:

(i) It holds that f is strongly F/pE, dq-measurable.

(ii) There exists a sequence gn : Ω Ñ E, n P N, of strongly F/pE, dq-measurablefunctions with the property that for all ω P Ω it holds that

lim supnÑ8


(iii) There exists a sequence gn : Ω Ñ E, n P N, of F/BpEq-simple functions withthe property that for all ω P Ω it holds that

lim supnÑ8


(iv) There exists a sequence gn : Ω Ñ E, n P N, of F/BpEq-simple functions withthe property that for all ω P Ω it holds that dpfpωq, gnpωqq P r0,8q, n P N,decreases monotonically to zero.

Proof of Theorem 2.3.10*. Throughout this proof assume w.l.o.g. thatE ‰ H. Clearly,it holds that ((iv) ñ (iii)) and ((iii) ñ (ii)). Lemma 2.3.9 shows that ((ii) ñ (i)).It thus remains to prove that ((i) ñ (iv)). For this let f : Ω Ñ E be a stronglyF/pE, dq-measurable function. The fact that f is strongly F/pE, dq-measurable en-sures that fpΩq is separable. Hence, there exists a sequence penqnPN Ď fpΩq ofelements in fpΩq with the property that

ten P fpΩq : n P Nu Ě fpΩq. (2.29)

In the next step let Ppe1,...,enq : E Ñ E, n P N, and gn : Ω Ñ E, n P N, be thefunctions with the property that for all x P E, n P N it holds that

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dpek,xq“distdptxu,te1,...,enuqu (2.30)

andgn “ Ppe1,...,enq ˝ f. (2.31)

Lemma 2.3.8 and the fact that f is F/BpEq-measurable implies that for all n P N itholds that gn is F/BpEq-measurable. In addition, by definition it holds for all n P Nthat gnpΩq Ď te1, . . . , enu is a finite set. We hence get that for all n P N it holds


that gn is an F/BpEq-simple function. Moreover, note that (2.22) in Lemma 2.3.8ensures that for all ω P Ω, n P N it holds that

dpfpωq, gnpωqq “ d`

fpωq, Ppe1,...,enqpfpωqq˘

“ distdptfpωqu, te1, . . . , enuq . (2.32)

This and the fact that @ω P Ω: distdpfpωq, te1, e2, . . . uq “ 0 imply that for all ω P Ω,n P N it holds that

dpfpωq, gnpωqq ě dpfpωq, gn`1pωqq and lim supnÑ8



2.3.5 Sums of strongly measurable functions

The next result, Corollary 2.3.11, shows that the sum of two strongly measurablemappings is again a strongly measurable mapping. Corollary 2.3.11 follows immedi-ately from Theorem 2.3.10.

Corollary 2.3.11 (*). Let pΩ,Fq be a measurable space, let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let f, g : Ω Ñ V be strongly F/pV, ¨V q-measurablemappings. Then f ` g : Ω Ñ V is strongly F/pV, ¨V q-measurable.

Class exercise 2.3.12 (*). The statement of Lemma 2.3.13 is in general not correct.Specify the mistake in the proof of Lemma 2.3.13.

Lemma 2.3.13 (*). Let pΩ,Fq be a measurable space, let pV, ¨V q be an R-Banachspace, let X, Y : Ω Ñ V be F/BpV q-measurable mappings, and let Z : Ω Ñ V be thefunction with the property that for all ω P Ω it holds that Zpωq “ Xpωq`Y pωq. Thenit holds that Z is F/BpV q-measurable.

2.4. CONTINUOUS FUNCTIONS 41

Proof of Lemma 2.3.13*. Throughout this proof let p : V ˆ V Ñ V be the functionwith the property that for all v, w P V it holds that

ppv, wq “ v ` w (2.34)

and let X : Ω Ñ V ˆ V be the function with the property that for all ω P Ω it holdsthat

Xpωq “ pXpωq, Y pωqq. (2.35)

Next observe that p : V ˆ V Ñ V is a continuous function from V ˆ V to V . This,in particular, ensures that p is a measurable function. Moreover, note that theassumption that X and Y are measurable ensures that X : Ω Ñ V ˆ V is alsomeasurable. This and the fact that p : V ˆ V Ñ V is measurable prove that thecomposition function p ˝ X : Ω Ñ V is measurable. Combining this with the factthat

Z “ p ˝X (2.36)

completes the proof of Lemma 2.3.13.

2.4 Continuous functions

2.4.1 Topological spaces

Definition 2.4.1 (Topology*). Let E be a set and let E Ď PpEq be a set

(i) such that H, E P E,

(ii) such that for all A Ď E it holds that YAPAA P E, and

(iii) such that for all A,B P E it holds that pAXBq P E.

Then and only then we say that E is a topology on E (we say that E is a topology).

Definition 2.4.2 (Topological space*). Let E be a set and let E be a topology on E.Then and only then we say that pE, Eq is a topological space.

Proposition 2.4.3 (Topology induced by a function*). Let E be a set, let T Ď Rbe a set, and let d : E ˆ E Ñ T be a function. Then it holds that the set

"

A P PpEq :´

@ v P A :“

D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰

¯

*

(2.37)

is a topology on E.


Proof of Proposition 2.4.3*. Throughout this proof let E Ď PpEq be the set givenby

E “"

A P PpEq :´

@ v P A :“


¯

*

. (2.38)

First, we observe thatH, E P E . Next we note that for all A Ď E and all v P rYAPAAsthere exists a set A P A and a real number ε P p0,8q such that v P A and such thattu P E : dpv, uq ă εu Ď A. In particular, this implies that for all A Ď E and allv P rYAPAAs there exists a real number ε P p0,8q such that tu P E : dpv, uq ă εu ĎrYAPAAs. Hence, we obtain that for all A Ď E it holds that

rYAPAAs P E . (2.39)

In the next step we observe that for all A,B P E and all v P pAXBq there existsreal numbers εA, εB P p0,8q such that

tu P E : dpv, uq ă εAu Ď A and tu P E : dpv, uq ă εBu Ď B. (2.40)

Hence, we obtain that for all A,B P E and all v P pAXBq there exists real numbersεA, εB P p0,8q such that

u P E : dpv, uq ă mintεA, εBu(

Ď pAXBq . (2.41)

This proves that for all A,B P E it holds that pAXBq P E . The proof of Proposi-tion 2.4.3 is thus completed.

Proposition 2.4.3 above ensures that the designation in the next definition isreasonable.

Definition 2.4.4 (Topology induced by a function*). Let E be a set, let T Ď R bea set, and let d : E ˆ E Ñ T be a function. Then we denote by τpdq Ď PpEq the setgiven by

τpdq “

"

A P PpEq :´

@ v P A :“


¯

*

(2.42)

and we call τpdq the topology induced by d.

Lemma 2.4.5 (Balls are open*). Let E be a set, let T Ď R be a set, let d : EÊ Ñ Tbe a function with the property that @x, y, z P E : dpx, zq ď dpx, yq ` dpy, zq, and letε P p0,8q, v P E. Then it holds that

tu P E : dpv, uq ă εu P τpdq. (2.43)


Proof of Lemma 2.4.5*. First, observe that for all x P tu P E : dpv, uq ă εu, y P tu PE : dpx, uq ă ε´ dpv, xqu it holds that

dpv, yq ď dpv, xq ` dpx, yq ă dpv, xq ` rε´ dpv, xqs “ ε. (2.44)

This ensures that for all x P tu P E : dpv, uq ă εu it holds that

tu P E : dpx, uq ă ε´ dpv, xqu Ď tu P E : dpv, uq ă εu (2.45)

Hence, we obtain that for all x P tu P E : dpv, uq ă εu there exists a real numberδ P p0,8q such that

tu P E : dpx, uq ă δu Ď tu P E : dpv, uq ă εu . (2.46)

This completes the proof of Lemma 2.4.5.

Proposition 2.4.6 (Convergence in the induced topology*). Let E be a set, letd : E ˆ E Ñ r0,8q be a function with the property that @x P E : dpx, xq “ 0 and@x, y, z P E : dpx, zq ď dpx, yq ` dpy, zq, and let e : N0 Ñ E be a function. Then itholds that

lim supnÑ8

d`

ep0q, epnq˘

“ 0 (2.47)

if and only if for every A P τpdq with ep0q P A there exists a natural number N P N

such that for all n P tN,N ` 1, . . . u it holds that epnq P A.

Proof of Proposition 2.4.6*. First of all, recall that lim supnÑ8 d`

ep0q, epnq˘

“ 0 ifand only if

@ ε P p0,8q : DN P N : @n P tN,N ` 1, . . . u : d`

ep0q, epnq˘

ă ε. (2.48)

Hence, we obtain that lim supnÑ8 d`

ep0q, epnq˘

“ 0 if and only if for all ε P p0,8qthere exists a natural number N P N such that for all n P tN,N ` 1, . . . u it holdsthat epnq P tu P E : dpep0q, uq ă εu. This and Lemma 2.4.5 complete the proof ofProposition 2.4.6.


2.4.2 Semi-metric spaces

Definition 2.4.7 (Semi-metric*). Let E be a set and let d : E ˆ E Ñ r0,8q be afunction with the property that for all x, y, z P E it holds that

(i) dpx, xq “ 0,

(ii) dpx, yq “ dpy, xq, and

(iii) dpx, zq ď dpx, yq ` dpy, zq.

Then and only then we say that d is a semi-metric on E (we say that d is a semi-metric).

Definition 2.4.8 (Semi-metric spaces*). Let E be a set and let d be a semi-metricon E. Then and only then we say that pE, dq is a semi-metric space.

Definition 2.4.9 (Globally bounded sets*). Let pE, dq be a semi-metric space andlet A Ď E be a set with the property that sup

`

t0u Y tdpa, bq : a, b P Au˘

ă 8. Thenand only then we say that A is d-globally bounded (we say that A is globally bounded).

Lemma 2.4.10 (Globally bounded sets*). Let pE, dq be a semi-metric space and letA Ď E be a non-empty set. Then the following three statements are equivalent:

(i) It holds that A is d-globally bounded.

(ii) It holds that @ e P E : supaPA dpa, eq ă 8.

(iii) It holds that D e P E : supaPA dpa, eq ă 8.

Proof of Lemma 2.4.10*. First, observe that for all e P E it holds that

supaPA

dpa, eq ď supa,bPA

dpa, bq. (2.49)

This ensures that ((i) ñ (ii)). Next note that the fact that E ‰ H shows that ((ii)ñ (iii)). It thus remains to prove that ((iii) ñ (i)). For this observe that the triangleinequality ensures that for all e P E it holds that

supa,bPA

dpa, bq ď supa,bPA

rdpa, eq ` dpe, bqs “ 2

„

supaPA

dpa, eq

. (2.50)

This establishes that ((iii) ñ (i)). The proof of Lemma 2.4.10 is thus completed.

Definition 2.4.11 (Globally bounded functions*). Let E be a set, let pF, dq be asemi-metric space, and let f : E Ñ F be a function with the property that fpEq is ad-globally bounded set. Then and only then we say that f is d-globally bounded (wesay that f is globally bounded).


2.4.3 Continuity properties of functions

Definition 2.4.12 (Continuous functions*). Let pE, Eq and pF,Fq be topologicalspaces. Then we denote by CpE,F q the set of all continuous functions from E to F .

Definition 2.4.13 (Uniformly continuous*). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function with the property that

@ ε P p0,8q : D δ P p0,8q : @x, y P E : ppdEpx, yq ă δq ñ pdF pfpxq, fpyqq ă εqq .(2.51)

Then and only then we say that f is uniformly continuous (we say that f is dE/dF -uniformly continuous).

Definition 2.4.14 (Holder continuous functions*). Let pE, dEq and pF, dF q be semi-metric spaces and let r P p0,8q. Then we denote by |¨|C0,rpE,F q : MpE,F q Ñ r0,8s

the function with the property that for all f PMpE,F q it holds that

|f |C0,rpE,F q “ sup

ˆ

t0u Y

"

dF pfpxq, fpyqq

|dEpx, yq|r P p0,8s : px, y P E, dF pfpxq, fpyqq ą 0q

*˙

(2.52)and we denote by C0,rpE,F q the set given by

C0,rpE,F q “

!

f PMpE,F q : |f |C0,rpE,F q ă 8

)

. (2.53)

Definition 2.4.15 (Holder continuous functions*). Let pE, dEq and pF, dF q be semi-metric spaces, let r P p0,8q, and let f P C0,rpE,F q. Then and only then we say thatf is r-Holder continuous (we say that f is dE/dF -r-Holder continuous)

Lemma 2.4.16. Let pV, ¨V q and pW, ¨W q be normed R-vector spaces, let U Ď V bean open set, let v P U , and let f : U Ñ W be a function which is Frechet differentiableat v. Then

lim suphŒ0

„

F pv ` hq ´ F pvqVhV

“ limεŒ0

suphPV zt0u,hV ďε

„

F pv ` hq ´ F pvqVhV

“ F 1pvqLpV,W q .

(2.54)


2.4.4 Modulus of continuity

Definition 2.4.17 (Modulus of continuity*). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function. Then we denote by wdE ,dFf : r0,8s Ñ r0,8sthe function with the property that for all h P r0,8s it holds that

wdE ,dFf phq “ sup´

t0u Y!

dF`

fpxq, fpyq˘

P r0,8q :“

x, y P E with dEpx, yq ď h‰

)¯

(2.55)and we call wdE ,dFf the dE/dF -modulus of continuity of f .

Lemma 2.4.18 (Properties of the modulus of continuity*). Let pE, dEq and pF, dF qbe semi-metric spaces and let f : E Ñ F be a function. Then

(i) it holds that wdE ,dFf is non-decreasing,

(ii) it holds that f is dE/dF -uniformly continuous if and only if limhŒ0wdE ,dFf phq “

0,

(iii) it holds that f is dF -globally bounded if and only if wdE ,dFf p8q ă 8,

(iv) it holds for all x, y P E that dF`

fpxq, fpyq˘

ď wdE ,dFf

`

dEpx, yq˘

, and

(v) it holds for all r P p0,8q that |f |C0,rpE,F q “ suphPp0,8q`

h´rwdE ,dFf phq˘

.

Proof of Lemma 2.4.18. First, observe that (i), (ii), and (iv) are an immediate con-sequence of the definition of wdE ,dFf . Next note that (iii) follows directly fromLemma 2.4.10. It thus remains to prove (v). For this observe that (iv) ensuresthat for all r P p0,8q, x, y P E with dEpx, yq ą 0 ă dF pfpxq, fpyqq it holds that

dF pfpxq, fpyqq

|dEpx, yq|rďwdE ,dFf pdEpx, yqq

|dEpx, yq|rď sup

hPp0,8q

˜

wdE ,dFf phq

hr

¸

. (2.56)

Next note that for all r P p0,8q, x, y P E with dEpx, yq “ 0 ă dF pfpxq, fpyqq it holdsthat

suphPp0,8q

˜

wdE ,dFf phq

hr

¸

ě suphPp0,8q

ˆ

dF pfpxq, fpyqq

hr

˙

“ 8. (2.57)

Combining this with (2.56) ensures that for all r P p0,8q it holds that |f |C0,rpE,F q ď

suphPp0,8qph´rwdE ,dFf phqq. Moreover, note that the fact that @ r P p0,8q, x, y P


E : dF pfpxq, fpyqq ď |f |C0,rpE,F q |dEpx, yq|r proves that

suphPp0,8q

˜

wdE ,dFf phq

hr

¸

ď suphPp0,8q

˜

|f |C0,rpE,F q hr

hr

¸

“ |f |C0,rpE,F q . (2.58)


The next result, Lemma 2.4.19, provides an upper bound for the modulus ofcontinuity of the point limit of a sequence of functions. Lemma 2.4.19 is, e.g., relatedto Item (i) in Corollary 2.10 in Cox et al. [2].

Lemma 2.4.19 (Convergence of the modulus of continuity). Let pE, dEq and pF, dF qbe semi-metric spaces and let fn : E Ñ F , n P N0, be functions which satisfy @ e PE : lim supnÑ8 dF pf0peq, fnpeqq “ 0. Then it holds for all h P r0,8s, r P p0,8q thatwdE ,dFf0

phq ď lim supnÑ8wdE ,dFfn

phq and |f0|C0,rpE,F q ď lim supnÑ8 |fn|C0,rpE,F q.

Proof of Lemma 2.4.19. Observe that for all h P r0,8s it holds that

wdE ,dFf0phq “ suppt0u Y tdF pf0pxq, f0pyqq P r0,8q : rx, y P E with dEpx, yq ď hsuq

ď sup

˜

t0u Y

#

lim supnÑ8“

dF pf0pxq, fnpxqq ` dF pfNpxq, fNpyqq

` dF pfnpyq, f0pyqq‰

P r0,8q : rx, y P E with dEpx, yq ď hs

+¸

ď sup

ˆ

t0u Y

"

lim supnÑ8

dF pfnpxq, fnpyqq P r0,8q : rx, y P E with dEpx, yq ď hs

*˙

ď lim supnÑ8wdE ,dFfn

phq.

(2.59)

This and Lemma 2.4.18 complete the proof of Lemma 2.4.19.

Exercise 2.4.20 (*). Give an example of a metric space pE, dq such that for allh P r0,8s it holds that

wd,didEphq “

#

0 : h P r0, 1q

1 : h P r1,8s. (2.60)

Prove that your metric space has the desired properties.


2.4.5 Extensions of uniformly continuous functions

Lemma 2.4.21 (Uniformly continuous functions*). Let pE, dEq and pF, dF q be semi-metric spaces, let f : E Ñ F be a uniformly continuous function, and let penqnPN Ď Ebe a Cauchy sequence. Then fpenq P F , n P N, is a Cauchy sequence too.

Proof of Lemma 2.4.21*. The assumption that penqnPN is a Cauchy sequence and theassumption that f is uniformly continuous imply that

limNÑ8

supn,mPtN,N`1,... u

dF pfpenq, fpemqq ď limNÑ8

supn,mPtN,N`1,... u

wdE ,dFf pdEpen, emqq

ď limNÑ8

wdE ,dFf

`

supn,mPtN,N`1,... u dEpen, emq˘

“ 0.(2.61)

This shows that fpenq P F , n P N, is a Cauchy sequence. The proof of Lemma 2.4.21is thus completed.

Proposition 2.4.22 (Extension of uniformly continuous functions*). Let pE, dEq bea semi-metric space, let pF, dF q be a complete semi-metric space, let A P PpEq, andlet f : AÑ F be a uniformly continuous function. Then

(i) there exists a unique f P CpA, F q with the property that f |A “ f ,

(ii) it holds for all h P r0,8s that

wdE ,dFf phq ď wdE ,dFf

phq ď limεŒ0

wdE ,dFf ph` εq, (2.62)

and

(iii) it holds that f is uniformly continuous.

Proof of Proposition 2.4.22*. The uniqueness of f is clear. It remains to provethe existence of a function f with the desired properties. For this observe thatfor all x P A and all penqnPN Ď A, penqnPN Ď A with lim supnÑ8 dEpen, xq “lim supnÑ8 dEpen, xq “ 0 it holds that

lim supnÑ8

dF`

fpenq, fpenq˘

ď lim supnÑ8

wdE ,dFf

`

dEpen, enq˘

“ 0. (2.63)

This, Lemma 2.4.21, and the assumption that pF, dF q is complete imply that thereexist a function f : AÑ F with the property that for all x P A and all penqnPN Ď Awith lim supnÑ8 dEpen, xq “ 0 it holds that

lim supnÑ8

dF`

fpenq, fpxq˘

“ 0. (2.64)


We observe that the continuity of f implies that for all x P A it holds that fpxq “fpxq. In the next step we show (2.62). The first inequality in (2.62) is clear. To provethe second inequality in (2.62) let h P r0,8s and let x0, y0 P A with dEpx0, y0q ď h.Then there exist sequences pxnqnPN Ď A and pynqnPN Ď A with the property thatlimnÑ8 xn “ x0 and limnÑ8 yn “ y0. This implies that for all ε P p0,8q it holds that

dF`

fpx0q, fpy0q˘

“ dF

´

limnÑ8

fpxnq, limnÑ8

fpynq¯

“ limnÑ8

dF pfpxnq, fpynqq

“ lim infnÑ8

dF pfpxnq, fpynqq ď lim infnÑ8

wdE ,dFf

`

dEpxn, ynq˘

ď wdE ,dFf

`

dEpx0, y0q ` ε˘

.

(2.65)

This proves the second inequality in (2.62). The second inequality in (2.62), inturn, shows that f is uniformly continuous. The proof of Proposition 2.4.22 is thuscompleted.

Exercise 2.4.23 (*). Specify a metric space pE, dEq, a complete metric space pF, dF q,a set A Ď E, and a uniformly continuous function f : A Ñ F such that the uniquefunction f P CpA, F q with f |A “ f satisfies wf ‰ wf (i.e., there exists an h P r0,8ssuch that wf phq ‰ wf phq).

Remark 2.4.24 (*). Let pE, dEq be a semi-metric space, let pF, dF q be a completesemi-metric space, let A Ď E be a subset of E, and let f : A Ñ F be a uniformlycontinuous function. Proposition 2.4.22 then proves that there exists a unique f PCpA, F q with f |A “ f . In the following we often write, for simplicity of presentation,f instead of f .

Definition 2.4.25 (*). LetK P tR,Cu, let V be aK-vector space, and let ¨V : V Ñr0,8q be a function such that

(i) it holds that 0V “ 0,

(ii) it holds for all v P V , λ P K that λvV “ |λ| vV , and

(iii) it holds for all v, w P V that v ` wV ď vV ` wV .

Then and only then we say that ¨V is a semi-norm on V .

Definition 2.4.26 (*). LetK P tR,Cu, let V be aK-vector space, and let ¨V : V Ñr0,8q be a semi-norm on V . Then and only then we say that pV, ¨V q is a semi-normed K-vector space.


Lemma 2.4.27 (*). Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be semi-normedK-vector spaces, and let A : V Ñ W be a linear mapping. Then A is continuous ifand only if A is uniformly continuous.

The proof of Lemma 2.4.27 is clear and therefore omitted.

Chapter 3

Linear functions and linear spaces

3.1 Sums over possibly uncountable index sets

3.1.1 Cofinal sequences

This section is based on Heuser [7, Satz 44.7].

Definition 3.1.1 (Cofinal sequence). Let pX,ĺq be a directed set and let x : NÑ Xbe a sequence with the property that for all y P X there exists a natural numberN P N such that for all n P tN,N ` 1, . . . u it holds that y ĺ xn. Then and only thenwe say that x is cofinal in pX,ĺq.

Example 3.1.2. Let x : N Ñ N be a function. Then it holds that x is cofinal in`

N, pN,N, tpa, bq : N2 : a ď buq˘

if and only if lim infnÑ8 xn “ 8.

There exists direct sets which do not admit any cofinal sequence. This is illus-trated in the next example; cf., e.g., Heuser [7, Exercise 6a].

Example 3.1.3. Let a P R, b P pa,8q, let X be the set given by

X “ tA P Ppra, bsq : #RpAq ă 8u , (3.1)

and let ĺ be the relation on X with the property that for all A,B P X it holds thatA ĺ B if and only if A Ď B. Then we note that for every sequence x : N Ñ X andevery t P

`

ra, bszpYnPNxnq˘

there exists no N P N such that ttu ĺ xN . Hence, thereexists no sequence x : NÑ X which is cofinal in pX,ĺq.

51

52 CHAPTER 3. LINEAR FUNCTIONS AND LINEAR SPACES

Proposition 3.1.4 (Convergence of cofinal sequences). Let pX,ĺq be a directed set,let pE, Eq be a topological space, let e P E, let φ : X Ñ E be a net which convergesto e, and let xn P X, n P N, be a cofinal sequence. Then it holds that the sequenceφpxnq, n P N, converges to e.

Proof of Proposition 3.1.4. Let U P E be an open set with the property that e P U .The assumption that φ converges to e ensures that there exists an element y P Xwith the property that for all z P X with y ĺ z it holds that

φpzq P U. (3.2)

In the next step we note that the assumption that xn, n P N, is cofinal implies thatthere exists a natural number N P N such that for all n P tN,N ` 1, . . . u it holdsthat

y ĺ xn. (3.3)

Combining (3.2) and (3.3) proves that for all n P tN,N ` 1, . . . u it holds that

φpxnq P U. (3.4)


Proposition 3.1.5 (Convergence of nets). Let pX,ĺq be a directed set, let xn P X,n P N, be a cofinal sequence, let pE, Eq be a topological space, let φ : X Ñ E be a net,and assume that for every cofinal seqence yn P X, n P N, it holds that φpynq P E,n P N, is a convergent sequence. Then there exists an element e P E such that φconverges to e and such that for every cofinal seqence yn P X, n P N, it holds thatφpynq P E, n P N, converges to e.

Proof of Proposition 3.1.5. Throughout this proof let p¨qb p¨q : MpN, Xq2 ÑMpN, Xqbe the mapping with the property that for all y, z : NÑ X it holds that

py b zqn “

#

zn2 : n is even

ypn`1q2 : n is odd(3.5)

Next observe that for all cofinal sequences y, z : NÑ X it holds that ybz is a cofinalsequence. By assumption we hence obtain that for all cofinal sequences y, z : NÑ Xit holds that

limnÑ8

φ pynq “ limnÑ8

φ`

py b zqn˘

“ limnÑ8

φ pznq . (3.6)

This proves that there exists an element e P E such that for every cofinal sequencey : NÑ X it holds that

limnÑ8

φpynq “ e. (3.7)

3.1. SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS 53

We now complete the proof of Proposition 3.1.5 by a contradiction, that is, weassume that φ does not converge to e. Hence, there exists an open set U P E withthe property that e P U and with the property that for all y P X there exists anelement z P X such that

y ĺ z and φpzq R U. (3.8)

This proves, in particular, that there exists a function z : N Ñ X such that for alln P N it holds that

xn ĺ zn and φpznq R U. (3.9)

The assumption that xn P X, n P N, is cofinal and the fact that @n P N : xn ĺ znproves that z : N Ñ X is cofinal too. This and (3.7) contradict to (3.9). The proofof Proposition 3.1.5 is thus completed.

Proposition 3.1.6 (Subsubsequence criterion). Let pE, Eq be a topological space andlet e : N0 Ñ E be a function. Then the following five statements are equivalent:

(i) It holds that en P E, n P N, converges in pE, Eq to e0.

(ii) For every strictly increasing function k : NÑ N there exists a strictly increasingfunction l : NÑ N such that ekplpnqq P E, n P N, converges in pE, Eq to e0.

(iii) For every strictly increasing function k : NÑ N there exists a function l : NÑN such that ekplpnqq P E, n P N, converges in pE, Eq to e0.

(iv) For every function k : NÑ N with lim infnÑ8 kpnq “ 8 there exists a functionl : N Ñ N with lim infnÑ8 lpnq “ 8 such that ekplpnqq P E, n P N, converges inpE, Eq to e0.

(v) For every function k : NÑ N with lim infnÑ8 kpnq “ 8 there exists a functionl : NÑ N such that ekplpnqq P E, n P N, converges in pE, Eq to e0.

Proof of Proposition 3.1.6. It is clear that ppiq ñ piiqq, ppiq ñ piiiqq, ppiq ñ pivqq,ppiq ñ pvqq, ppiiq ñ piiiqq, ppivq ñ pvqq, and pvq ñ piiiq. It is thus sufficient to provethat ppiiiq ñ piqq to complete the proof of Proposition 3.1.6. We prove ppiiiq ñ piqqby a contradiction, that is, we assume

`

piiiq^ p piqq˘

in the following. Observe thatp piqq ensures that there exists a set A P E with e0 P A and #Nptn P N : en R Auq “8. Therefore, there exists a strictly increasing function k : NÑ N such that for alln P N it holds that

ekpnq R A. (3.10)


Next note that piiiq assures that there exists a function l : NÑ N such that ekplpnqq PE, n P N, converges in pE, Eq to e0. This implies that there exists a natural numberN P N such that for all n P tN,N`1, . . . u it holds that ekplpnqq P A. In particular, weobtain that ekplpNqq P A. This contradicts to (3.10). The proof of Proposition 3.1.6is thus completed.

3.1.2 Sums over possibly uncountable index sets

Definition 3.1.7 (*). Let A be a set and let f : AÑ r0,8s be a function. Then wedenote by

ř

aPA fpaq the extended real number in r0,8s with the property that

ÿ

aPA

fpaq “

ż

A

fpaq#Apdaq. (3.11)

Another way to define the sum in (3.11) above is to employ the concept of a net.This is illustrated in the following example.

Example 3.1.8 (Sums through nets*). Let A be a set, let f : AÑ r0,8s be a func-tion, and let φ : tx P PpAq : #Apxq ă 8u Ñ r0,8s be the function with the propertythat for all finite subsets x Ď A of A it holds that

φpxq “ÿ

aPx

fpaq. (3.12)

Then it holds that the pair

ptx P PpAq : #Apxq ă 8u ,Ďq (3.13)

is a directed set and it holds that φ is a net which converges toř

aPA fpaq.

3.1.3 Fubini for sums

Definition 3.1.9 (*). Let d P N, A P BpRdq. Then we denote by BorelA : BpAq Ñr0,8s the Lebesgue-Borel measure on A.

Let us illustrate Definition 3.1.9 through a simple example. Note that for alla, b, α, β P R with a ď α ď β ď b it holds that

Borelra,bsprα, βsq “ β ´ α. (3.14)

Definition 3.1.10 (Finiteness of measures*). Let µ be a measure with the propertythat impµq Ď R. Then and only then we say that µ is finite.

3.1. SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS 55

Definition 3.1.11 (Sigma-finiteness of measures*). Let pΩ,F , µq be a measure spacewith the property that there exists a sequence An P F , n P N, of sets such thatYnPNAn “ Ω and such that for all n P N it holds that µpAnq ă 8. Then and onlythen we say that µ is sigma-finite.

Example 3.1.12 (On the sigma-finitness in the Fubini’s theorem*). It holds thatż

r0,1s

ż

r0,1s

1txupyq#RpdyqBorelRpdxq “

ż 1

0

ÿ

yPr0,1s

1txupyq dx “

ż 1

0

#Rptxuq dx “ 1,

ż

r0,1s

ż

r0,1s

1txupyq BorelRpdxq#Rpdyq “ÿ

yPr0,1s

ż 1

0

1txupyq dx “ÿ

yPr0,1s

BorelRptyuq “ 0.

(3.15)

Lemma 3.1.13 (*). Let A be a set and let f : A Ñ r0,8s be a function withř

aPA fpaq ă 8. Then it holds that the set f´1pp0,8sq is at most countable.

Proof of Lemma 3.1.13*. Monotonicity proves that for all ε P p0,8q it holds that

#A

`

f´1prε,8sq

˘

¨ ε ďÿ

aPA

fpaq ă 8. (3.16)

This shows that for all n P N it holds that the set f´1`

r1n,8s˘

is finite. This impliesthat the set

f´1`

p0,8s˘

“ YnPNf´1`

r1n,8s˘

(3.17)

is at most countable. The proof of Lemma 3.1.13 is thus completed.

Lemma 3.1.14 (Fubini for sums*). Let A and B be sets and let f : AˆB Ñ r0,8sbe a function. Then

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

bPB

ÿ

aPA

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq. (3.18)

Proof of Lemma 3.1.14*. W.l.o.g. we assume that pAˆBq ‰ H is a non-empty set.We prove that

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq. (3.19)

Clearly, (3.19) implies (3.18). To prove (3.19), we distinguish between several cases.In the first case we assume that

ÿ

pa,bqPAˆB

fpa, bq ă 8. (3.20)


This assumption together with Lemma 3.1.13 implies that there exists a sequencepan, bnq P Aˆ B, n P N, such that for all px, yq P pAˆ Bqztpan, bnq : n P Nu it holdsthat fpx, yq “ 0. The theorem of Fubini hence proves that

ÿ

pa,bqPAˆB

fpa, bq “ÿ

pa,bqPtan : nPNuˆtbn : nPNu

fpa, bq

“

ż

fpa, bq #tan : nPNuˆtbn : nPNupda, dbq

“

ż ż

fpa, bq #tbn : nPNupdbq #tan : nPNupdaq

“ÿ

aPtan : nPNu

ÿ

bPtbn : nPNu

fpa, bq “ÿ

aPA

ÿ

bPB

fpa, bq.

(3.21)

This finishes the proof of (3.19) in the case

ÿ

pa,bqPAˆB

fpa, bq ă 8. (3.22)

In the second case we assume that

ÿ

aPA

ÿ

bPB

fpa, bq ă 8. (3.23)

Lemma 3.1.13 implies then that there exists an at most countable set A Ď A suchthat for all a P AzA it holds that

ř

bPB fpa, bq “ 0. Moreover, again Lemma 3.1.13implies that there exist at most countable sets Ba Ď B, a P A, such that for alla P A, b P BzBa it holds that fpa, bq “ 0. The theorem of Fubini hence shows that

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

aPA

ÿ

bPBa

fpa, bq “ÿ

aPA

ÿ

bPpYaPABaq

fpa, bq

“ÿ

pa,bqPAˆpYaPABaq

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq.(3.24)


3.2. SETS OF INTEGRABLE FUNCTIONS 57

3.2 Sets of integrable functions

3.2.1 Lp-sets of measurable functions for p P r0,8q

Definition 3.2.1 (Lp-sets for p P r0,8q*). LetK P tR,Cu, let pΩ,A, µq be a measurespace, let q P p0,8q, and let pV, ¨V q be a normed K-vector space. Then we denote

by L0pµ; ¨V q the set given by

L0pµ; ¨V q “MpA,BpV qq, (3.25)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that

for all f P L0pµ; ¨V q it holds that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

, (3.26)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (3.27)

Definition 3.2.2 (Equivalence classes*). Let pΩ,F , µq be a measure space, let pS,Sqbe a measurable space, let R be a set, and let f : Ω Ñ R be a function. Then we denoteby rf sµ,S the set given by

rf sµ,S “!

g PMpF ,Sq :“

DA P F :`

µpAq “ 0 and tω P Ω: fpωq ‰ gpωqu Ď A˘‰

)

.

(3.28)

Definition 3.2.3 (Lp-sets for p P r0,8q*). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,F , µq be a measure space, and let pV, ¨V q be a normed K-vector space. Then

we denote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “

rf sµ,BpV q ĎMpF ,BpV qq : f P Lppµ; ¨V q

(

(3.29)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property

that for all f P L0pµ; ¨V q it holds that

rf sµ,BpV qLqpµ;¨V q“ fLqpµ;¨V q

. (3.30)


3.2.2 Lp-spaces of strongly measurable functions for p P r0,8q

Definition 3.2.4 (Lp-spaces for p P r0,8q*). Let K P tR,Cu, q P p0,8q, letpΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Then wedenote by L0pµ; ¨V q the set given by

L0pµ; ¨V q “ tf PMpΩ, V q : f is strongly A/pV, ¨V q-measurableu, (3.31)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that

for all f P L0pµ; ¨V q it holds that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

P r0,8s, (3.32)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (3.33)

Corollary 2.3.11 proves, in the setting of Definition 3.2.4, that for all p P r0,8qit holds that Lppµ; ¨V q is a K-vector space.

Definition 3.2.5 (Equivalence classes of strongly measurable functions*). Let K P

tR,Cu, let pV, ¨V q be a normed K-vector space, let pΩ,F , µq be a measure space,let R Ď V be a set, and let f : Ω Ñ R be a function. Then we denote by tfuµ,¨V

theset given by

tfuµ,¨V

“

!

g P L0pµ; ¨V q :

“

DA P F :`

µpAq “ 0 and tω P Ω: fpωq ‰ gpωqu Ď A˘‰

)

.

(3.34)

Definition 3.2.6 (Lp-spaces for p P r0,8q*). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Thenwe denote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “!

tfuµ,¨VĎ L0

pµ; ¨V q : f P Lppµ; ¨V q

)

(3.35)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property

that for all f P L0pµ; ¨V q it holds that

tfuµ,¨V Lqpµ;¨V q“ fLqpµ;¨V q

. (3.36)

3.3. LINEAR SPACES 59

Lemma 3.2.7 (Theorem of Fischer-Riesz for equivalence classes of strongly mea-surable functions*). Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a measure space,and let pV, ¨V q be a K-Banach space. Then it holds that Lppµ; ¨V q is a K-Banachspace.

Lemma 3.2.8 (*). Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a finite measurespace, and let pV, ¨V q be a normed K-vector space. Then it holds that the set

tfuµ,¨V : f is an F/BpV q-simple function(

(3.37)

is dense in Lppµ; ¨V q.

Proof of Lemma 3.2.8*. Throughout this proof let f P Lppµ; ¨V q. Theorem 2.3.10proves that there exists a sequence gn : Ω Ñ V , n P N, of F/BpV q-simple functionswith the property that for all ω P Ω it holds that fpωq ´ gnpωqV , n P N, decreasesmonotonically to zero. Lebesgue’s theorem of dominated convergence hence showsthat

lim supnÑ8

f ´ gnLppµ;¨V q“ lim sup

nÑ8

„ż

Ω

fpωq ´ gnpωqpV µpdωq

1p

“ 0. (3.38)


3.3 Linear spaces

We first recall some notions regarding linear spaces (also known as vector spaces).

Definition 3.3.1 (Span in a vector space*). Let K be a field, let V be a K-vectorspace, and let A Ď V . Then we denote by spanV pAq the set with the property that

spanV pAq “ t0u Y tλ1a1 ` . . .` λnan P V : n P N, a1, . . . , an P A, λ1, . . . , λn P Ku .(3.39)

Definition 3.3.2 (Generating system*). Let V be a vector space and let A Ď V bea set with the property that spanV pAq “ V . Then and only then we say that A is agenerating system in V .

Definition 3.3.3 (Counting measure on a set*). Let A be a set. Then we denote by#A : PpAq Ñ r0,8s the counting measure on A.


Definition 3.3.4 (Linearly independent*). Let K be a field, let V be a K-vectorspace, and let A Ď V be a set with the property that for all n P N, λ1, . . . , λn P K,a1, . . . , an P A with #Apta1, . . . , anuq “ n and λ1a1 ` . . .` λnan “ 0 it holds that

λ1 “ . . . “ λn “ 0. (3.40)

Then and only then we say that A is linearly independent in V .

Definition 3.3.5 (Basis of a vector space*). Let K be a field, let V be a K-vectorspace, and let A be a linearly independent generating system in V . Then and onlythen we say that A is a Hamel basis of V .

Theorem 3.3.6 (Every vector space has a basis*). Let K be a field and let V be aK-vector space. Then there exists a Hamel basis of V .

3.4 Hilbert spaces

In the next step we recall some notions regarding Hilbert spaces.

3.4.1 Orthonormal bases

Definition 3.4.1 (Orthogonal*). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbert space and letA Ď H be a set with the property that for all a, b P A with a ‰ b it holds that

〈a, b〉H “ 0. (3.41)

Then and only then we say that A is orthogonal in pH, 〈¨, ¨〉H , ¨Hq.

Definition 3.4.2 (Orthonormal*). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H be a set which is orthogonal in pH, 〈¨, ¨〉H , ¨Hq and which satisfies for alla P A that

aH “ 1. (3.42)

Then and only then we say that A is orthonormal in H (we say that A is orthonormalin pH, 〈¨, ¨〉H , ¨Hq).

Definition 3.4.3 (Completeness*). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H be a set with the property that

spanHpAq “ H. (3.43)

Then and only then we say that A is complete in H (we say that A is complete inpH, 〈¨, ¨〉H , ¨Hq).

3.4. HILBERT SPACES 61

Definition 3.4.4 (Orthonormal basis*). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert spaceand let A Ď H be a set which is complete and orthonormal in pH, 〈¨, ¨〉H , ¨Hq. Thenand only then we say that A is an orthonormal basis of H (we say that A is anorthonormal basis of pH, 〈¨, ¨〉H , ¨Hq).

Class exercise 3.4.5 (*). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and let A Ď Hbe an orthonormal basis of H. Is A then a Hamel basis of H?

Theorem 3.4.6 (Every Hilbert space has an orthonormal basis*). Let pH, 〈¨, ¨〉H , ¨Hqbe a Hilbert space. Then there exists an orthonormal basis A Ď H of H.

Proposition 3.4.7 (A characterization for separable Hilbert spaces*). Let K P

tR,Cu and let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space. Then H is separable if andonly if there exists an at most countable orthonormal basis A Ď H of H.

Proof of Proposition 3.4.7. W.l.o.g. we assume that H ‰ t0u. If A Ď H is an atmost countable orthonormal basis of H, then the set

tλ1a1 ` . . .` λnan : n P N, a1, . . . , an P A, λ1, . . . , λn P tx P K : Repxq, Impxq P Quu(3.44)

is an at most countable dense subset of H. This proves the “ð” direction in thestatement of Proposition 3.4.7. The “ñ” direction in in the statement of Proposi-tion 3.4.7 follows from an application of the Gram-Schmidt process.

Definition 3.4.8 (Absolute value functions*). We denote by |¨|C

: C Ñ r0,8q,|¨| : C Ñ r0,8q, and |¨|

R: R Ñ r0,8q the functions with the property that for all

a, b P R it holds that

|a` bi|C “ |a` bi| “?a2 ` b2 and |a|R “ |a|C “ |a|. (3.45)

Example 3.4.9 (*). Let er : RÑ R, r P R, be the functions with the property thatfor all r, x P R it holds that

erpxq “

#

1 : x “ r

0 : x ‰ r. (3.46)

Then note that tteru P L2p#R; |¨|

Rq : r P Ru is an orthonormal basis of the Hilbert

space L2p#R; |¨|Rq. Proposition 3.4.7 hence proves that the Hilbert space L2p#R; |¨|

Rq

is not separable.


Lemma 3.4.10. Let n P N, x “ px1, . . . , xnq P Rn, p P r1,8q. Then

xLpp#t1,2,...,nu;|¨|Rq“ p|x1|

p` . . .` |xn|

pq1pď |x1| ` . . .` |xn| “ xL1p#t1,2,...,nu;|¨|Rq

.

(3.47)

Proof of Lemma 3.4.10. Throughout this proof let e1, . . . , en P Rn be the vectors

given by e1 “ p1, 0, . . . , 0q, e2 “ p0, 1, 0, . . . , 0q, . . . , en “ p0, . . . , 0, 1q. Next observethat the triangle inequality implies that

xLpp#t1,2,...,nu;|¨|Rq“

›

›

›

›

›

nÿ

k“1

xkek

›

›

›

›

›

Lpp#t1,2,...,nu;|¨|Rq

ď

nÿ

k“1

xkekLpp#t1,2,...,nu;|¨|Rq

“

nÿ

k“1

|xk| ekLpp#t1,2,...,nu;|¨|Rq“

nÿ

k“1

|xk| “ xL1p#t1,2,...,nu;|¨|Rq.

(3.48)


3.4.2 Best approximations and projections in Hilbert spaces

Theorem 3.4.11 (Best approximations and projections in Hilbert spaces*). LetpH, 〈¨, ¨〉H , ¨Hq be a Hilbert space and let U Ď H be a closed subspace of H. Thenthere exists a unique P P LpHq with the property that for all v P H it holds thatP pHq Ď U and

P pvq ´ vH “ infwPU

w ´ vH . (3.49)

Definition 3.4.12 (Projection in Hilbert spaces*). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbertspace and let U Ď H be a closed subspace of H. Then we denote by PH,U P LpHqthe unique bounded linear operator from H to H which satisfies for all v P H thatPH,UpHq Ď U and

PH,Upvq ´ vH “ infwPU

w ´ vH (3.50)

and we call PH,U the projection of H on U .


3.4.3 Examples of orthonormal bases

3.4.3.1 Elementary properties of trigonometric functions

Lemma 3.4.13 (Real and imaginary part of product of complex numbers). For allz1, z2 P C it holds that

Repz1 ¨ z2q “ Repz1q ¨Repz2q ´ Impz1q ¨ Impz2q , (3.51)

Impz1 ¨ z2q “ Repz1q ¨ Impz2q ` Impz1q ¨Repz2q . (3.52)

The proof of Lemma 3.4.13 is clear. The next lemma presents a well-knownidentity for the difference of two arguments of the cosine function.

Lemma 3.4.14. For all x, y P R it holds that

cosp2xq ´ cosp2yq “ 2 sinpy ´ xq sinpy ` xq. (3.53)

Proof of Lemma 3.4.14. Throughout this proof let ϕ, ϕ0,1, ϕ1,1 P C8pR2,Rq be the

functions with the property that for all x, y P R it holds that

ϕpx, yq “ cosp2xq ´ cosp2yq ´ 2 sinpy ´ xq sinpy ` xq,

ϕ0,1px, yq “`

B

Byϕ˘

px, yq, ϕ1,1px, yq “`

B2

BxByϕ˘

px, yq.(3.54)

Next observe that for all x, y P R it holds that

ϕpx, yq “ ϕpx, 0q `

ż y

0

ϕ0,1px, sq ds

“ ϕpx, 0q `

ż y

0

„

ϕ0,1p0, sq `

ż x

0

ϕ1,1pr, sq dr

ds

“ ϕpx, 0q `

ż y

0

ϕ0,1p0, sq ds`

ż y

0

ż x

0

ϕ1,1pr, sq dr ds.

(3.55)

Moreover, observe that for all x, y P R it holds that

ϕ0,1px, yq “ 2 sinp2yq ´ 2 cospy ´ xq sinpy ` xq ´ 2 sinpy ´ xq cospy ` xq, (3.56)

ϕ0,1p0, yq “ 2 sinp2yq ´ 4 sinpyq cospyq, (3.57)


ϕ1,1px, yq “ ´2 sinpy ´ xq sinpy ` xq ´ 2 cospy ´ xq cospy ` xq

` 2 cospy ´ xq cospy ` xq ` 2 sinpy ´ xq sinpy ` xq “ 0.(3.58)

Equation (3.57) and Lemma 3.4.13 imply that for all y P R it holds that

ϕ0,1p0, yq “ 2 psinpy ` yq ´ cospyq sinpyq ´ sinpyq cospyqq

“ 2`

Im`

eiy ¨ eiy˘

´Re`

eiy˘

¨ Im`

eiy˘

´ Im`

eiy˘

¨Re`

eiy˘˘

“ 0.(3.59)

This, (3.58) and (3.55) prove that for all x, y P R it holds that

ϕpx, yq “ ϕpx, 0q “ cosp2xq ´ 1´ 2 sinp´xq sinpxq “ cosp2xq ´ 1` 2 |sinpxq|2

“ cosp2xq ` |sinpxq|2 ´ |cospxq|2

“ Re`

eix ¨ eix˘

´“

Re`

eix˘

¨Re`

eix˘

´ Im`

eix˘

¨ Im`

eix˘‰

.

(3.60)

This and Lemma 3.4.13 imply that for all x, y P R it holds that ϕpx, yq “ 0. Theproof of Lemma 3.4.14 is thus completed.


3.4.3.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq

Proposition 3.4.15 (Examples of orthonormal sets). It holds

(i) that the set!

“

p?

2 sinpnπxqqxPp0,1q‰

Borelp0,1q,BpRqP L2

pBorelp0,1q; |¨|Rq : n P N)

(3.61)

is orthonormal in L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacian with Dirich-let boundary conditions),

(ii) that the set

1(

Y

!

“

p?

2 cospnπxqqxPp0,1q‰

Borelp0,1q,BpRqP L2


(3.62)

is orthonormal in L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacian with Neu-mann boundary conditions),

(iii) that the set

1(

Y

!

“

p?

2 sinp2nπxqqxPp0,1q‰

Borelp0,1q,BpRqP L2


Y

!

“

p?

2 cosp2nπxqqxPp0,1q‰

Borelp0,1q,BpRqP L2


(3.63)

is orthonormal in L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacian with pe-riodic boundary conditions), and

(iv) that the set!

“

p?

2 sinppn´ 12qπxqqxPp0,1q‰

Borelp0,1q,BpRqP L2


(3.64)

is orthonormal in L2pBorelp0,1q; |¨|Rq (Eigenfunctions associated to a standardBrownian motion r0, 1s).

Proof of Proposition 3.4.15. First of all, note that for all n P Rzt0u it holds that

ż 1

0

1 ¨?

2 cospnπxq dx “?

2

ż 1

0

cospnπxq dx “?

2

„

sinpnπxq

nπ

x“1

x“0

“?

2

„

sinpnπq ´ sinp0q

nπ

“ 0.

(3.65)


Next observe that integration by parts proves that for all n P Rzt0u, m P R it holdsthat

ż 1

0

sinpnπxq sinpmπxq dx

“

„

´ cospnπxq sinpmπxq

nπ

x“1

x“0

`mπ

nπ

ż 1

0

cospnπxq cospmπxq dx

“m

n

ż 1

0

cospnπxq cospmπxq dx.

(3.66)

In addition, observe that (3.51) together with (3.65) ensures that for all n,m P R

with n`m ‰ 0 it holds thatż 1

0

sinpnπxq sinpmπxq dx “

ż 1

0

Im`

einπx˘

¨ Im`

eimπx˘

dx

“

ż 1

0

Re`

einπx˘

¨Re`

eimπx˘

´Re`

einπx ¨ eimπx˘

dx

“

ż 1

0

cospnπxq cospmπxq dx´

ż 1

0

Re`

eipn`mqπx˘

dx

“

ż 1

0

cospnπxq cospmπxq dx.

(3.67)

Putting (3.67) into (3.66) proves that for all n P Rzt0u, m P Rztn,ńu it holds that

ż 1

0


ż 1

0

cospnπxq cospmπxq dx “ 0. (3.68)

This, (3.65), and (3.67) ensure for all n P R, m P Rztn,ńu that

ż 1

0


ż 1

0

cospnπxq cospmπxq dx “ 0. (3.69)

In addition, observe that (3.67) implies that for all n P Rzt0u it holds that

1 “

ż 1

0

|sinpnπxq|2 ` |cospnπxq|2looooooooooooooomooooooooooooooon

“1

dx “ 2

ż 1

0

|sinpnπxq|2 dx. (3.70)

This and (3.67) show that for all n P Rzt0u it holds that

ż 1

0

ˇ

ˇ

?2 sinpnπxq

ˇ

ˇ

2dx “

ż 1

0

ˇ

ˇ

?2 cospnπxq

ˇ

ˇ

2dx “ 1. (3.71)


Combining this with (3.69) assures that for all n,m P N it holds that

ż 1

0

?2 sinpnπxq ¨

?2 sinpmπxq dx “

ż 1

0

?2 cospnπxq ¨

?2 cospmπxq dx

“

#

0 : n ‰ m

1 : n “ m.

(3.72)

This and (3.70) prove that the set (3.61) is an orthonormal set in L2pBorelp0,1q; |¨|Rq.Next we combine (3.65) and (3.72) to obtain that the set (3.62) is an orthonormal setin L2pBorelp0,1q; |¨|Rq. Furthermore, we observe that (3.52) ensures for all n,m P R

with n`m ‰ 0 it holds that

ż 1

0

sinpnπxq cospmπxq dx`

ż 1

0

cospnπxq sinpmπxq dx “

ż 1

0

Im`

einπx ¨ eimπx˘

dx

“

ż 1

0

sinppn`mqπxq dx “

„

´ cosppn`mqπxq

pn`mqπ

x“1

x“0

“1´ cosppn`mqπq

pn`mqπ.

(3.73)

This assures that for all n,m P N it holds thatş1

0sinp2nπxq cosp2mπxq dx “ 0. This

together with (i) and (ii) proves (iii). Moreover, we observe that (3.51) and (3.65)imply that for all n,m P N it holds that

ż 1

0

sinppn´ 12qπxq sinppm´ 12qπxq dx

“

ż 1

0

Im`

eipn´12qπx˘

¨ Im`

eipm´12qπx˘

dx

“

ż 1

0

Re`

eipn´12qπx˘

¨Re`

eipm´12qπx˘

dx´Re`

eipn´12qπx¨ eipm´12qπx

˘

dx

“

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx´Re

ˆż 1

0

eipn`m´1qπx dx

˙

“

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx.

(3.74)


Furthermore, integration by parts proves that for all n,m P N it holds thatż 1

0

sinppn´ 12qπxq sinppm´ 12qπxq dx

“

„

´ cosppn´ 12qπxq sinppm´ 12qπxq

pn´ 12qπ

x“1

x“0

`pm´ 12qπ

pn´ 12qπ

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx

“pm´ 12qπ

pn´ 12qπ

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx.

(3.75)

As above we combine (3.74) and (3.75) to obtain that the set (3.64) is orthonormalin L2pBorelp0,1q; |¨|Rq.

Definition 3.4.16 (Hausdorff). Let pE, Eq be a topological space. Then pE, Eq iscalled Hausdorff if and only if for all a, b P E with a ‰ b there exists A,B P E witha P A, b P B, and AXB “ H.

Definition 3.4.17 (Hausdorff space). E is called Hausdorff space if and only if (Eis a topological space and E is Hausdorff).

Theorem 3.4.18 (Stone-Weierstrass). Let K P tR,Cu, let pE, Eq be a Hausdorffspace, and let A Ď CpE,Kq be a subalgebra of CpE,Kq such that

(i) @ v P A : v P A (A is a sub-*-algebra of CpE,Kq),

(ii) 1 P A (A is a sub-*-algebra of CpE,Kq with 1) and

(iii) @x, y P E, x ‰ y : D v P A : vpxq ‰ vpyq (A seperates points).

Then A is dense in CpE,Kq.

A proof of Theorem 3.4.18 in German language can, for example, be found inHeuser [6]. We illustrate Theorem 3.4.18 by the following result.

Proposition 3.4.19. Let S “ tpa, bq P R2 : a2 ` b2 “ 1u Ď R2, let arg : S Ñ r0, 2πqbe the function with the property that for all x P S it holds that

`

cospargpxqq, sinpargpxqq˘

“ x, (3.76)

and let A Ď CpS,Cq be the set given by

A “ď

NPN

"ˆ

Nř

n“Ń

an ein argpxq

˙

xPS

P CpS,Cq : aŃ , a1Ń , . . . , aN P C

*

. (3.77)

Then A is dense in CpS,Cq.


Proof of Proposition 3.4.19. We prove Proposition 3.4.19 through an application ofTheorem 3.4.18. For this we note that for all v P A it holds that v P A. Moreover,note that

`

ei 0 argpxq˘

xPS“`

e0˘

xPS“ 1 P A. (3.78)

Furthermore, observe that for all n,m P Z, x P S it holds that

ein argpxq¨ eim argpxq

“ ei pn`mq argpxq (3.79)

This ensures that A is a subalgebra of CpS,Cq. In order to apply Theorem 3.4.18, itremains to verify that A separates points. For this observe that pei argpxqqxPS P CpS,Cqand that for all x, y P S with argpxq ą argpyq it holds that

ei argpxq

ei argpyq“ eipargpxqárgpyqq

‰ 1. (3.80)

This ensures that for all x, y P S with x ‰ y it holds that

ei argpxq‰ ei argpyq. (3.81)

We can thus apply Theorem 3.4.18 to obtain that A is dense in CpS,Cq. The proofof Proposition 3.4.19 is thus completed.

Corollary 3.4.20. It holds that the set

!

`

1?2πeinx

˘

xPp0,2πqP L2

pBorelp0,2πq; |¨|Cq : n P Z)

(3.82)

is an orthonormal basis of L2pBorelp0,2πq; |¨|Cq.

Proof of Corollary 3.4.20. Observe that for all n,m P Z it holds that

ż 2π

0

1?2πeinx 1?

2πeimx dx “

1

2π

ż 2π

0

eipmńqx dx

“

$

&

%

”

12πipmńq

eipmńqxıx“2π

x“0“ 0 : m ‰ n

1 : m “ n.

(3.83)

This proves that the set `

1?2πeinx

˘

xPp0,2πqP L2pBorelp0,2πq; |¨|Cq : n P Z

(

is orthonor-

mal in L2pBorelp0,2πq; |¨|Cq. Next we denote by

CP pr0, 2πs,Cq “ tf P Cpr0, 2πs,Cq : fp0q “ fp2πqu (3.84)


the set of all 2π-periodic continuous functions from r0, 2πs to C. Proposition 3.4.19implies that

span!

`

1?2πeinx

˘

xPp0,2πq: n P Z

)CP pr0,2πs,Cq

“ CP pr0, 2πs,Cq. (3.85)

This implies that

span!

`

1?2πeinx

˘

xPp0,2πq: n P Z

)L2pBorelp0,2πq;|¨|Cq

Ě CP pr0, 2πs,CqL2pBorelp0,2πq;|¨|Cq

“ L2pBorelp0,2πq; |¨|Cq.

(3.86)


Exercise 3.4.21. Prove that the sets!

“

1‰

Borelp0,1q,BpRq

)

Y

!

“

pcospnxqqxPp0,πq‰

Borelp0,1q,BpRqP L2

pBorelp0,πq; |¨|Rq : n P N)

(3.87)and

!

“

psinpnxqqxPp0,πq‰

Borelp0,1q,BpRqP L2

pBorelp0,πq; |¨|Rq : n P N)

(3.88)

are orthonormal bases of L2pBorelp0,πq; |¨|Rq.


Proposition 3.4.22 (Examples of orthonormal bases*). It holds

(i) that the set!

“

p?


Borelp0,1q,BpRq: n P N

)

(3.89)

is an orthonormal basis of L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacianwith Dirichlet boundary conditions),

(ii) that the set!

“

1‰

Borelp0,1q,BpRq

)

Y

!

“

p?

2 cospnπxqqxPp0,1q‰


)

(3.90)

is an orthonormal basis of L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacianwith Neumann boundary conditions),

(iii) that the set!

“

1‰

Borelp0,1q,BpRq

)

Y

!

“

p?

2 sinp2nπxqqxPp0,1q‰


)

Y

!

“

p?

2 cosp2nπxqqxPp0,1q‰


) (3.91)

is an orthonormal basis of L2pBorelp0,1q; |¨|Rq (Eigenfunctions of the Laplacianwith periodic boundary conditions), and

(iv) that the set!

“

p?

2 sinppn´ 12qπxqqxPp0,1q‰


)

(3.92)

is an orthonormal basis of L2pBorelp0,1q; |¨|Rq (Eigenfunctions associated to astandard Brownian motion r0, 1s).

3.4.3.3 Transformations of orthonormal bases

Exercise 3.4.23 (Transformation of orthonormal bases*). Let a, α P R, b P pa,8q,β P pα,8q, let en PMpBppa, bqq,BpRqq, n P N, be functions with the property thatthe set trensBorelpa,bq,BpRq : n P Nu is an orthonormal basis of L2pBorelpa,bq; |¨|Rq, andlet fn : pα, βq Ñ R, n P N, be the functions with the property that for all n P N,x P pα, βq it holds that

fnpxq “

d

pb´ aq

pβ ´ αqen

ˆ

px´ αq pb´ aq

pβ ´ αq` a

˙

. (3.93)

Prove that the set trfnsBorelpα,βq,BpRq P L2pBorelpα,βq; |¨|Rq : n P Nu is an orthonormal

basis of L2pBorelpα,βq; |¨|Rq.


3.5 Linear functions

In the section we particularly follow the presentations in Werner [23].

Definition 3.5.1 (Linear operators*). Let K be a field, let V1 and V2 be K-vectorspaces, and let A : V1 Ñ V2 be a function/operator1 with the property that for allv, w P V1, λ P K it holds that

A pλv ` wq “ λAv ` Aw. (3.94)

Then and only then we say that A is called is linear (we say that A is K-linear).

Definition 3.5.2 (*). Let K be a field and let V1 and V2 be K-vector spaces. Thenwe denote by LinpV1, V2q the set given by

LinpV1, V2q “ tA PMpV1, V2q : A is linearu (3.95)

(the set of all linear functions from V1 to V2/the set of all linear operators from V1

to V2).

Definition 3.5.3 (Linear operators on a vector space*). Let K be a field, let V1 andV2 be K-vector spaces, let U Ď V1 be a vector subspace of V1, and let A P LinpU, V2q.Then and only then we say that A is a linear operator from U on V1 to V2 (we saythat A is a linear operator on V1 to V2).

Definition 3.5.4 (Set of linear operators on a vector space*). Let K be a field andlet V1 and V2 be K-vector spaces. Then we denote by LpV1, V2q the set given by

LpV1, V2q “ď

UĎV1 is a vectorsubspace of V1

LinpU, V2q (3.96)

(the set of linear operators on V1 to V2).

Definition 3.5.5 (Point spectrum of a linear operator*). Let K P tR,Cu, let V bea K-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Then we denoteby σP pAq the set given by

σP pAq “!

λ P K :`

λ´ A : DpAq Ñ V is not injective˘

)

(3.97)

and we call σP pAq the point spectrum of A (the set of eigenvalues of A).

1A function from some possibly infinite dimensional normed vector space into some possiblyinfinite dimensional normed vector space is also often referred as an “operator”. We will also oftenuse this convention in the remainder of these lecture notes.

3.5. LINEAR FUNCTIONS 73

Definition 3.5.6 (Symmetric linear operators*). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator. Then we saythat A is symmetric if and only if it holds for all v, w P DpAq that

〈Av,w〉H “ 〈v, Aw〉H . (3.98)

Definition 3.5.7 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space,and let A : DpAq Ď H Ñ H be a linear operator. Then we say that A is nonnegativeif and only if it holds for all v P DpAq that

〈v, Av〉H P r0,8q. (3.99)

Definition 3.5.8 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a linear operator. Then we say that A is strictly positiveif and only if it holds for all v P DpAqzt0u that

〈v,Av〉H P p0,8q. (3.100)

Definition 3.5.9 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space,and let A : DpAq Ď H Ñ H be a linear operator. Then we say that A is nonpositiveif and only if it holds for all v P DpAq that

〈v, Av〉H P p´8, 0s. (3.101)

Definition 3.5.10 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space,and let A : DpAq Ď H Ñ H be a linear operator. Then we say that A is strictlynegative if and only if it holds for all v P DpAqzt0u that

〈v, Av〉H P p´8, 0q. (3.102)

3.5.1 Continuous linear functions on normed vector spaces

Definition 3.5.11 (Bounded linear functions/bounded linear operators*). Let K P

tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be normed K-vector spaces. Then we denoteby LpV1, V2q the set given by

LpV1, V2q “ LinpV1, V2q X CpV1, V2q (3.103)

(the set of all continuous linear operators from V1 to V2) and we denote by ¨LpV1,V2q :

LpV1, V2q Ñ r0,8q the function with the property that for all A P LpV1, V2q it holdsthat

ALpV1,V2q “ supvPV1zt0u

„

AvV2vV1

. (3.104)


Definition 3.5.12 (*). Let K P tR,Cu and let pV, ¨V q be a normed K-vector space.Then we denote by LpV1q the set given by

LpV1q “ LpV1, V1q (3.105)

(the set of all continuous linear operators from V1 to V1) and we denote by ¨LpV1q :

LpV1q Ñ r0,8q the function with the property that for all A P LpV1q it holds that

ALpV1q “ ALpV1,V1q . (3.106)

Lemma 3.5.13 (Completeness of the space of bounded linear operators*). Let K PtR,Cu, let pV1, ¨V1q be a normed K-vector space, and let pV2, ¨V2q be a K-Banachspace. Then pLpV1, V2q, ¨LpV1,V2qq is a K-Banach space.

Definition 3.5.14 (Topological dual space*). Let K P tR,Cu and let pV, ¨V q be anormed K-vector space. Then we denote by pV 1, ¨V 1q the K-Banach space given by

pV 1, ¨V 1q “ pLpV,Kq, ¨LpV,Kqq. (3.107)

See Reed & Simon [19] and, e.g., Prevot & Rockner [18] for the next results.

Theorem 3.5.15 (Square root of a nonnegative and symmetric bounded linearoperator*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A PLpHq be nonnegative and symmetric. Then there exists a unique symmetric andnonnegative S P LpHq with the property that S2 “ A.

Definition 3.5.16 (Square root of a nonnegative and symmetric bounded linearoperator*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A PLpHq be nonnegative and symmetric. Then we denote by A12 P LpHq the uniquesymmetric and nonnegative bounded linear operator with the property that pA12q2 “

A.

Lemma 3.5.17 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A P LpHq. Then A˚A is nonnegative and symmetric.

Proof of Lemma 3.5.17*. Note that for all v P H it holds that

〈v, A˚Av〉H “ 〈Av,Av〉H “ 〈A˚Av, v〉H “ 〈Av,Av〉H “ Av

2H ě 0. (3.108)



3.5.2 Nuclear operators on Banach spaces

3.5.2.1 Definition of Nuclear operators

Definition 3.5.18 (Rank-1 operators*). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe normed K-vector spaces, let v P V 1, and let w P W . Then we denote by

`

w bv˘

: V Ñ W the function with the property that for all u P V it holds that

pw b vqpuq “ vpuqw. (3.109)

Definition 3.5.19 (Nuclear operator*). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe K-Banach spaces, and let A : V Ñ W be a K-linear operator with the property thatthere exist a sequence pvnqnPN Ď V 1 of elements in V 1 and a sequence pwnqnPN Ď Wof elements in W such that for all x P V it holds that

8ÿ

n“1

vnV 1 wnW ă 8 and Ax “8ÿ

n“1

pwn b vnqpxq “8ÿ

n“1

vnpxqwn. (3.110)

Then and only then we say that A is nuclear.

Condition (3.110) says something about how good a linear operator can be ap-proximated through sums of linear operators with one-dimensional images. In addi-tion, condition (3.110) asserts that a nuclear operator can be decomposed into rankone operators in the sense of (3.110). The identity in (3.110) is also referred to as anuclear represenation of a nuclear operator.

Definition 3.5.20 (The normed vector space of nuclear operators*). Let K P tR,Cuand let pV, ¨V q and pW, ¨W q be K-Banach spaces. Then we denote by L1pV,W qthe set given by

L1pV,W q “ tA P LinpV,W q : A is nuclearu (3.111)

(the set of all nuclear operators from V to W ) and we denote by ¨L1pV,W q: L1pV,W q Ñ

r0,8q the function with the property that for all A P L1pV,W q it holds that

AL1pV,W q“ inf

#

a P r0,8q :

„

D pvnqnPN Ď V 1 : D pwnqnPN Ď W :

´

a “ř8

n“1 vnV 1 wnW ă 8 and @x P V : Ax “ř8

n“1pwn b vnqpxq¯

+

. (3.112)


Definition 3.5.21 (*). Let K P tR,Cu and let pV, ¨V q be a K-Banach space.Then we denote by L1pV q the set given by L1pV q “ L1pV, V q and we denote by¨L1pV q

: L1pV q Ñ r0,8q the mapping with the property that for all A P L1pV q it

holds that AL1pV q“ AL1pV,V q

.

3.5.2.2 Relation of bounded linear operators and nuclear operators

Lemma 3.5.22 (*). Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaces,and let A P L1pV,W q. Then A P LpV,W q and it holds that

ALpV,W q ď AL1pV,W q. (3.113)

Proof of Lemma 3.5.22. Throughout this proof let ε P p0,8q be a real number. Theassumption that A P L1pV,W q ensures that there exist sequences pvnqnPN Ď V 1 andpwnqnPN Ď W such that

8ÿ

n“1

vnV 1 wnW ď ε` AL1pV,W qď ε`

8ÿ

n“1

vnV 1 wnW ă 8 (3.114)

and such that for all x P V it holds that

Ax “8ÿ

n“1

vnpxqwn. (3.115)

Next note that for all x P V it holds that

AxW “

›

›

›

›

›

8ÿ

n“1

vnpxqwn

›

›

›

›

›

W

ď

8ÿ

n“1

vnpxqwnW “

8ÿ

n“1

|vnpxq| wnW

ď

8ÿ

n“1

vnLpV,Rq xV wnW “

«

8ÿ

n“1

vnV 1 wnW

ff

xV

ď

”

AL1pV,W q` ε

ı

xV .

(3.116)

This proves that A P LpV,W q and that

ALpV,W q ď AL1pV,W q` ε. (3.117)

As ε P p0,8q was arbitrary, the proof of Lemma 3.5.22 is completed.

Lemma 3.5.22, in particular, proves that in the setting of Lemma 3.5.22 it holdsthat

pL1pV,W q, ¨L1pV,W qq Ď pLpV,W q, ¨LpV,W qq (3.118)

continuously.


3.5.2.3 Structure of the space of nuclear operators

Lemma 3.5.23 (*). Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be K-Banachspaces. Then the pair pL1pV,W q, ¨L1pV,W q

q is a normed K-vector space.

Proof of Lemma 3.5.23. Lemma 3.5.22 implies that for all A P L1pV,W q it holdsthat AL1pV,W q

“ 0 if and only if A “ 0. Furthermore, it is clear that for all

A P L1pV,W q, λ P K it holds that λ ¨ A P L1pV,W q and that

λ ¨ AL1pV,W q“ |λ| ¨ AL1pV,W q

. (3.119)

It thus remains to prove that the sum of two nuclear operators from V to W is againa nuclear operator from V to W and that the triangle inequality holds. For this letA1, A2 P L1pV,W q, ε P p0,8q be arbitrary and let vin P V

1, n P N, i P t1, 2u, andwin P W , n P N, i P t1, 2u, satisfy that for all i P t1, 2u it holds that

8ÿ

n“1

›

›vin›

›

V 1

›

›win›

›

Wď ε` AiL1pV,W q

ď ε`8ÿ

n“1

›

›vin›

›

V 1

›

›win›

›

Wă 8 (3.120)

and that for all x P V , i P t1, 2u it holds that

Aix “8ÿ

n“1

vinpxqwin. (3.121)

This implies that8ÿ

n“1

2ÿ

i“1

›

›vin›

›

V 1

›

›win›

›

Wă 8 (3.122)

and that for all x P V it holds that

pA1 ` A2qx “8ÿ

n“1

2ÿ

i“1

vinpxqwin

“ v11pxqw

11 ` v

21pxqw

21 ` v

12pxqw

12 ` v

22pxqw

22 ` v

13pxqw

13 ` v

23pxqw

23 ` . . . .

(3.123)

Hence, we obtain that A1 ` A2 P L1pV,W q and that

A1 ` A2L1pV,W qď

8ÿ

n“1

2ÿ

i“1

›

›vin›

›

V 1

›

›win›

›

Wď A1L1pV,W q

` A2L1pV,W q` 2ε. (3.124)

As ε P p0,8q was arbitrary, the proof of Lemma 3.5.23 is completed.


3.5.2.4 Ideal property of the set of nuclear operators

Proposition 3.5.24. Let K P tR,Cu, let pV0, ¨V0q, pV1, ¨V1q, pW0, ¨W0q and

pW1, ¨W1q be K-Banach spaces, and let A P L1pV1,W1q, B1 P LpW1,W0q, B2 P

LpV0, V1q. Then it holds that B1AB2 P L1pV0,W0q and it holds that

B1AB2L1pV0,W0qď B1LpW1,W0q

AL1pV1,W1qB2LpV1,V0q . (3.125)

Proof of Proposition 3.5.24. Let ε P p0,8q be arbitrary. The assumption that A PL1pV1,W1q ensures that there exist pvnqnPN Ď pV1q

1 and pwnqnPN Ď W1 such that

8ÿ

n“1

vnV 11wnW1

ď ε` AL1pV1,W1qď ε`

8ÿ

n“1

vnV 11wnW1

ă 8 (3.126)

and such that for all x P V1 it holds that

Ax “8ÿ

n“1

vnpxqwn. (3.127)

Inequality (3.126) implies that

8ÿ

n“1

vnpB2p¨qqV 10B1pwnqW0

ď

8ÿ

n“1

vnV 11B2LpV0,V1q B1LpW1,W0q

wnW1

ď B2LpV0,V1q B1LpW1,W0q

”

ε` AL1pV1,W1q

ı

(3.128)

and equation (3.127) ensures that for all x P V0 it holds that

pB1AB2q pxq “8ÿ

n“1

vnpB2pxqqB1pwnq. (3.129)

This implies that B1AB2 is a nuclear operator from V0 to W0 and that

B1AB2L1pV0,W0qď B2LpV0,V1q B1LpW1,W0q

”

ε` AL1pV1,W1q

ı

. (3.130)

As ε P p0,8q was arbitrary, the proof of Proposition 3.5.24 is completed.


3.5.2.5 Characterization of nuclear operators

The next simple lemma gives a characterization for nuclear operators and is animmediate consequence of the definition of a nuclear operator.

Lemma 3.5.25 (*). Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaceswith V ‰ t0u and W ‰ t0u, and let A P LpV,W q. Then the following three state-ments are equivalent:

(i) It holds that A P L1pV,W q.

(ii) There exist panqnPN Ď R, pvnqnPN Ď V 1, pwnqnPN Ď W such that for all n P Nit holds that vnV 1 “ wnW “ 1, such that

ř8

n“1 |an| ă 8 and such that forall x P V it holds that

Ax “8ÿ

n“1

an vnpxqwn. (3.131)

(iii) There exist panqnPN Ď r0,8q, pvnqnPN Ď V 1, pwnqnPN Ď W such that for alln P N it holds that vnV 1 “ wnW “ 1, such that

ř8

n“1 an ă 8 and such thatfor all x P V it holds that

Ax “8ÿ

n“1

an vnpxqwn. (3.132)

3.5.3 Hilbert-Schmidt operators on Hilbert spaces

Definition 3.5.26. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q

be K-Hilbert spaces, and let A P LpH1, H2q be a bounded linear operator with theproperty that there exist an orthonormal basis B Ď H1 of H1 such that

ÿ

bPB

Ab2H2ă 8. (3.133)

Then and only then we say that A is Hilbert-Schmidt.

3.5.3.1 Independence of the orthonormal basis

Lemma 3.5.27 (*). Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q

be K-Hilbert spaces, let B1 Ď H1 be an orthonormal basis of H1, let B2 Ď H2 be anorthonormal basis of H2, and let A P LpH1, H2q. Then

ÿ

bPB1

Ab2H2“

ÿ

bPB2

A˚b2H1. (3.134)


Proof of Lemma 3.5.27*. Observe thatÿ

bPB1

Ab2H2“

ÿ

bPB1

ÿ

bPB2

ˇ

ˇxb, AbyH2

ˇ

ˇ

2“

ÿ

bPB1

ÿ

bPB2

ˇ

ˇxA˚b, byH1

ˇ

ˇ

2

“ÿ

bPB2

ÿ

bPB1

ˇ

ˇxA˚b, byH1

ˇ

ˇ

2“

ÿ

bPB2

A˚b2H1.

(3.135)


Lemma 3.5.28 (Independence of the orthonormal bases*). Let K P tR,Cu, letpH1, 〈¨, ¨〉H1

, ¨H1q and pH2, 〈¨, ¨〉H2

, ¨H2q be K-Hilbert spaces, let B1 Ď H1 and

B2 Ď H1 be orthonormal bases of H1, and let A P LpH1, H2q. Thenÿ

bPB1

Ab2H2“

ÿ

bPB2

Ab2H2. (3.136)

Proof of Lemma 3.5.28*. Theorem 3.4.6 implies that there exists an orthonormalbasis B Ď H2 of H2. Lemma 3.5.27 then implies that

ÿ

bPB1

Ab2H2“

ÿ

bPB

A˚b2H1“

ÿ

bPB2

Ab2H2. (3.137)


The next result, Corollary 3.5.29, gives a characterization of Hilbert-Schmidtoperators and follows immediately from Lemma 3.5.28 above.

Corollary 3.5.29 (*). LetK P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q

be K-Hilbert spaces, and let A P LpH1, H2q. Then A is a Hilbert-Schmidt operator ifand only if for every orthonormal basis B Ď H1 of H1 it holds that

ÿ

bPB

Ab2H1ă 8. (3.138)

3.5.3.2 The Hilbert space of Hilbert-Schmidt operators

Definition 3.5.30 (The space of Hilbert-Schmidt operators*). Let K P tR,Cu andlet pH1, 〈¨, ¨〉H1

, ¨H1q and pH2, 〈¨, ¨〉H2

, ¨H2q be K-Hilbert spaces. Then we denote

by L2pH1, H2q and HSpH1, H2q the sets given by

L2pH1, H2q “ HSpH1, H2q “ tA P LpH1, H2q : A is Hilbert-Schmidtu (3.139)

(the set of all Hilbert-Schmidt operators from H1 to H2).


In the next definition, Definition 3.5.31, we introduce a norm on a space of Hilbert-Schmidt operators. Lemma 3.5.28 above ensures that (3.140) in Definition 3.5.31 doesindeed make sense.

Definition 3.5.31 (A norm on the space of Hilbert-Schmidt operators*). Let K P

tR,Cu and let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q be K-Hilbert spaces. Thenwe denote by ¨L2pH1,H2q

“ ¨HSpH1,H2q: L2pH1, H2q Ñ r0,8q the function with the

property that for all A P L2pH1, H2q and all orthonormal bases B Ď H1 of H1 it holdsthat

AL2pH1,H2q“ AHSpH1,H2q

“

«

ÿ

bPB

Ab2H2

ff12

. (3.140)

3.5.3.3 Hilbert-Schmidt embeddings

Lemma 3.5.32 (Hilbert-Schmidt embeddings). Let rn P Rzt0u, n P N, be realnumbers with

ř8

n“11

|rn|2ă 8, let pH, 〈¨, ¨〉H , ¨Hq and pH, 〈¨, ¨〉H , ¨Hq be R-Hilbert

spaces with H Ď H continuously, and let en P H, n P N, be an orthonormal basis ofH which satisfies that rnen, n P N, is an orthonormal basis of H. Then it holds forall v, w P H that

〈v, w〉H “8ÿ

n“1

〈en, v〉H 〈en, w〉H|rn|

2 . (3.141)

Proof. Note that for all n P N, v, w P H it holds that

〈en, v〉H “

⟨en,

8ÿ

m“1

〈em, v〉H em

⟩H

“

8ÿ

m“1

〈em, v〉H 〈en, em〉H

“ 〈en, v〉H 〈en, en〉H “ 〈en, v〉H en2H “

〈en, v〉H|rn|2

¨ rnen2H “

〈en, v〉H|rn|2

.

(3.142)

This implies for all v, w P H that

〈v, w〉H “8ÿ

n“1

〈rnen, v〉H 〈rnen, w〉H “8ÿ

n“1

〈en, v〉H 〈en, w〉H|rn|

2 . (3.143)



3.6 Diagonal linear operators on Hilbert spaces

Definition 3.6.1 (Diagonal linear operators*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator with theproperty that there exists an orthonormal basis B Ď H of H and a function λ : BÑ K

such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(3.144)

and such that for all v P DpAq it holds that

Av “ÿ

bPB

λb 〈b, v〉H b. (3.145)

Then and only then we say that A is a diagonal linear operator.

Definition 3.6.2 (Densely defined*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator. Then we say that A

is a densely defined if and only if DpAqH“ H.

Class exercise 3.6.3 (Diagonal operators are densely defined*). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Prove that A is densely defined.

Exercise 3.6.4 (Symmetry of diagonal linear operators*). Let pH, 〈¨, ¨〉H , ¨Hq bean R-Hilbert space and let A : DpAq Ď H Ñ H be a diagonal linear operator. Provethat A is symmetric.

Exercise 3.6.5 (The point spectrum of a diagonal linear operator*). LetK P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linear operator,let B Ď H be an orthonormal basis of H, and let λ : BÑ K be a function such thatfor all v P DpAq it holds that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(3.146)

and Av “ř

bPB λb 〈b, v〉H b. Prove that σP pAq “ impλq.

3.6. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 83

Proposition 3.6.6 (Regularity of diagonal linear operators*). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linear operator,let B Ď H be an orthonormal basis of H, and let λ : BÑ K be a function such thatfor all v P DpAq it holds that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(3.147)

and Av “ř

bPB λb 〈b, v〉H b. Then

(i) A P LpHq if and only if λ P L8p#B; |¨|Kq (ô λ is a bounded function ô

impλq “ σP pAq is a bounded set) and in that case it holds that ALpHq “

λL8p#B;|¨|Kq“ supbPB |λb|,

(ii) A P L2pHq “ HSpHq if and only if λ P L2p#B; |¨|Kq (ô

ř

bPB |λb|2ă 8) and

in that case it holds that AL2pHq“ λL2p#B;|¨|

Kq““ř

bPB |λb|2‰12

, and

(iii) A P L1pHq if and only if λ P L1p#B; |¨|Kq (ô

ř

bPB |λb| ă 8) and in that caseit holds that AL1pHq

“ λL1p#B;|¨|Kq“ř

bPB |λb|.

Class exercise 3.6.7 (*). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let b : NÑ Hbe an injective mapping such that bpNq “ tb1, b2, . . . u Ď H is an orthonormal basisof H, let Ar : DpArq Ď H Ñ H, r P R, be linear operators with the property that forall r P R, v P DpArq it holds that DpArq “

v P H :ř8

n“1 n2r |〈bn, v〉H |

2ă 8

(

and

Arv “8ÿ

n“1

nr 〈bn, v〉H bn. (3.148)

Specify three sets S1 Ď R, S2 Ď R, and S3 Ď R of real numbers

(i) such that for all r P R it holds that Ar P HSpHq if and only if r P S1,

(ii) such that for all r P R it holds that Ar P LpHq if and only if r P S2, and

(iii) such that for all r P R it holds that Ar P L1pHq if and only if r P S3.

Definition 3.6.8 (Closedness of a linear operator*). Let K P tR,Cu, let pV, ¨V q bea normed K-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Then we

say that A is linearly closed (we say that A is closed) if and only if GraphpAqVˆV

“

GraphpAq.


Proposition 3.6.9 (Closedness of diagonal linear operators*). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then A is linearly closed.

Proof of Proposition 3.6.9. Throughout this proof let x, y P H, let vn P DpAq, n P N,be a sequence which satisfies lim supnÑ8 x´vnH “ lim supnÑ8 yÁvnH “ 0, letB Ď H be an orthonormal basis of H, and let λ : BÑ K be a function such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(3.149)


Av “ÿ

bPB

λb 〈b, v〉H b. (3.150)

Fatou’s lemma then proves that

0 “ lim supnÑ8

y ´ Avn2H “ lim sup

nÑ8

«

ÿ

bPB

|〈b, y〉H ´ λb 〈b, vn〉H |2

ff

“ lim infnÑ8

«

ÿ

bPB

|〈b, y〉H ´ λb 〈b, vn〉H |2

ff

ěÿ

bPB

”

lim infnÑ8

|〈b, y〉H ´ λb 〈b, vn〉H |2ı

“ÿ

bPB

|〈b, y〉H ´ λb 〈b, x〉H |2 .

(3.151)

This and the fact thatř

bPB |〈b, y〉H |2ă 8 ensure that x P DpAq and Ax “ y. The

proof of Proposition 3.6.9 is thus completed.

3.6.1 Laplace operators on bounded domains

3.6.1.1 Laplace operators with Dirichlet boundary conditions

In this section we give functional analytic descriptions of Laplace operators with suit-able boundary conditions and thereby present a few important examples of diagonallinear operators.


Definition 3.6.10 (Laplace operator with Dirichlet boundary conditions*). Let en PL2pBorelp0,1q; |¨|Rq, n P N, satisfy for all n P N that en “ rp

?2 sinpnπxqqxPp0,1qsBorelp0,1q,BpRq,

and let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator withthe property that for all v P DpAq it holds that

DpAq “

#

v P L2pBorelp0,1q; |¨|Rq :

8ÿ

n“1

n4ˇ

ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq

ˇ

ˇ

2

Ră 8

+

(3.152)

and

Av “8ÿ

n“1

´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.153)

Then we refer to A as the Laplace operator with Dirichlet boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.6.11 (*). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq bethe Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq. Then

(i) it holds that A is a diagonal linear operator,

(ii) it holds that σP pAq “ t´π2, ´4π2, ´9π2, ´16π2, . . . u,

(iii) it holds that

DpAq “ H2pp0, 1q;Rq

loooooomoooooon

Sobolev space

XH10 pp0, 1q;Rq

loooooomoooooon

Sobolev space

“

$

’

’

’

&

’

’

’

%

v P H2pp0, 1q;Rq : lim

xŒ0vpxq

looomooon

“vp0`q

“ limxÕ1

vpxqlooomooon

“vp1´q

“ 0

,

/

/

/

.

/

/

/

-

,

(3.154)

and

(iv) it holds for all v P DpAq that Av “ v2.


3.6.1.2 Laplace operators with Neumann boundary conditions

Definition 3.6.12 (Laplace operator with Neumann boundary conditions). Leten P L2pBorelp0,1q; |¨|Rq, n P N0, satisfy that for all n P N and Borelp0,1q-almost

all x P p0, 1q it holds that e0pxq “ 1 and enpxq “?

2 cospnπxq, and let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with the property that

DpAq “

#

v P L2pBorelp0,1q; |¨|Rq :

8ÿ

n“1

n4ˇ


ˇ

ˇ

2

Ră 8

+

(3.155)

and with the property that for all v P DpAq it holds that

Av “8ÿ

n“0

´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.156)

Then we refer to A as the Laplace operator with Neumann boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.6.13. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Neumann boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´π2, ´4π2,´9π2, ´16π2, . . . u, it holds that

DpAq “

$

’

’

’

&

’

’

’

%


xŒ0v1pxq

looomooon

“v1p0`q

“ limxÕ1

v1pxqlooomooon

“v1p1´q

“ 0

,

/

/

/

.

/

/

/

-

, (3.157)

and it holds for all v P DpAq that Av “ v2.


3.6.1.3 Laplace operators with periodic boundary conditions

Definition 3.6.14 (Laplace operator with periodic boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P Z, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q

it holds that e0pxq “ 1, enpxq “?

2 sinp2nπxq and eńpxq “?

2 cosp2nπxq, and letA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with theproperty that

DpAq “

#

v P H :ÿ

nPZ

n4ˇ


ˇ

ˇ

2

Ră 8

+

(3.158)

and with the property that for all v P DpAq it holds that

Av “ÿ

nPZ

´4π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.159)

Then we refer to A as the Laplace operator with periodic boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.6.15. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with periodic boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´4¨π2, ´4¨22 ¨π2,´4 ¨ 32 ¨ π2, ´4 ¨ 42 ¨ π2, . . . u, it holds that

DpAq “ H2P pp0, 1q;Rq

loooooomoooooon

Sobolev space

“

$

’

’

’

&

’

’

’

%


xŒ0vpxq

looomooon

“vp0`q

“ limxÕ1

vpxqlooomooon

“vp1´q

, limxŒ0

v1pxqlooomooon

“v1p0`q

“ limxÕ1

v1pxqlooomooon

“v1p1´q

,

/

/

/

.

/

/

/

-

,

(3.160)

and it holds for all v P DpAq that Av “ v2.


3.6.2 Spectral decomposition for a diagonal linear operator

Proposition 3.6.16 (The eigenspaces of diagonal linear operators*). LetK P tR,Cube a field, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linearoperator, let B Ď H be an orthonormal basis of H, and let λ : BÑ K be a functionsuch that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(3.161)


Av “ÿ

bPB

λb 〈b, v〉H b. (3.162)

Then

(i) it holds for all µ P σP pAq “ Impλq that

Kernpµ´ Aq “ spantb P B : λb “ µu “ spanpλ´1ptµuqq, (3.163)

”

spantb P B : λb “ µuı

k

”

spantb P B : λb ‰ µuı

“ H (3.164)

and

(ii) it holds for all µ1, µ2 P σP pAq, v1 P Kernpµ1 ´ Aq, v2 P Kernpµ2 ´ Aq withµ1 ‰ µ2 that 〈v1, v2〉H “ 0.

Proof of Proposition 3.6.16*. Let µ P σP pAq be arbitrary. We first prove that

Kernpµ´ Aq Ď spantb P B : λb “ µu. (3.165)

For this observe that for all

v P Kernpµ´ Aq “ tw P Dpµ´ Aq “ DpAq : pµ´ Aqw “ 0u (3.166)

it holds that

0 “ 02H “ pµ´ Aqv2H “

ÿ

bPB

|pµ´ λbq 〈b, v〉H |2“

ÿ

bPB

|µ´ λb|2|〈b, v〉H |

2

“ÿ

bPB,λb‰µ

|µ´ λb|2|〈b, v〉H |

2 .(3.167)


Hence, we obtain that for all v P Kernpµ ´ Aq and all b P B with λb ‰ µ it holdsthat 〈b, v〉H “ 0. This implies that for all v P Kernpµ´ Aq it holds that

v P”

spantb P B : λb ‰ µuıK

. (3.168)

This and the identity

”

spantb P B : λb ‰ µuı

k

”

spantb P B : λb “ µuı

“ H (3.169)

prove that (3.165) is indeed fufilled. Next we prove that

Kernpµ´ Aq Ě spantb P B : λb “ µu. (3.170)

For this observe that for all

v P spantb P B : λb “ µu “”


(3.171)

it holds that

ÿ

bPB

|λb 〈b, v〉H |2“

ÿ

bPBλb“µ

|λb 〈b, v〉H |2“ |µ|2

ÿ

bPBλb“µ

|〈b, v〉H |2“ |µ|2 v2H ă 8. (3.172)

This shows that

spantb P B : λb “ µu “”


Ď DpAq “ Dpµ´ Aq. (3.173)

Next note that for all

v P spantb P B : λb “ µu “”


(3.174)

it holds that

pµ´ Aq v “ÿ

bPB

pµ´ λbq 〈b, v〉H b “ÿ

bPB,λb“µ

pµ´ λbq 〈b, v〉H b “ 0. (3.175)



The next result, Theorem 3.6.17, establishes a spectral decomposition for diagonallinear operators. It follows immediately from Proposition 3.6.16 above.

Theorem 3.6.17 (Spectral decomposition for diagonal linear operators*). Let K P

tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator. Then it holds that

DpAq “

$

&

%

v P H :ÿ

λPσP pAq

|λ|2›

›PKernpλÁq,Hpvq›

›

2

Hă 8

,

.

-

(3.176)

and it holds for all v P DpAq that

Av “ÿ

λPσP pAq

λ ¨ PKernpλÁq,Hpvq. (3.177)

Exercise 3.6.18 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a diagonal linear operator. Prove that A is symmetric ifand only if σP pAq Ď R.

Proposition 3.6.19 (Orthonormal basis of eigenfunctions). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a diagonal linearoperator, let B Ď DpAq be an orthonormal basis of H, and let λ : BÑ K be a functionwhich satisfies for all b P B that Ab “ λbb. Then

(i) it holds that DpAq “

v P H :ř

bPB |λb 〈b, v〉H |2ă 8

(

and

(ii) it holds for all v P DpAq that Av “ř

bPB λb 〈b, v〉H b.

Proof of Proposition 3.6.19. Throughout this proof let B : DpBq Ď H Ñ H be thelinear operator with the property that

DpBq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(3.178)

and with the property that for all v P DpBq it holds that

Bv “ÿ

bPB

λb 〈b, v〉H b. (3.179)


Next observe that the assumptions that A is a linear operator, that B Ď DpAq, andthat @ b P B : Ab “ λbb imply that for all v P spanpBq it holds that

Av “ Bv. (3.180)

Moreover, note that Proposition 3.6.9 implies that A is a closed linear operator.This, Lemma 3.1.13, and (3.180) ensure that for all v P DpBq it holds that v P DpAqand Av “ Bv. Hence, we obtain that

GraphpBq Ď GraphpAq. (3.181)

Furthermore, observe that the assumptions that B Ď DpAq and that @ b P B : Ab “λbb and Exercise 3.6.5 assure that

Impλq “ σP pBq Ď σP pAq. (3.182)

Item (ii) in Proposition 3.6.16 proves that for all µ P σP pAqz Impλq, v P KernpµÁq,b P B it holds that

〈v, b〉H “ 0. (3.183)

This and the assumption that B is an orthonormal basis ensure that for all µ PσP pAqz Impλq, v P Kernpµ ´ Aq it holds that v “ 0. Hence, we obtain for allµ P σP pAqz Impλq that Kernpµ ´ Aq “ t0u. This implies that σP pAqz Impλq “ H.Combining this with (3.182) proves that

σP pAq “ Impλq “ σP pBq. (3.184)

Next observe that the assumption that @ b P B : Ab “ λbb implies that for allµ P σP pAq it holds that λ´1ptµuq “ tb P B : λb “ µu Ď Kernpµ ´ Aq. Item (i) inProposition 3.6.16 hence ensures for all µ P σP pAq that

spantb P B : λb “ µu Ď Kernpµ´ Aq. (3.185)

This, (3.184), and again Item (i) in Proposition 3.6.16 prove for all µ P σP pAq “σP pBq “ Impλq that

Kernpµ´Bq “ spantb P B : λb “ µu Ď Kernpµ´ Aq. (3.186)

Item (ii) in Proposition 3.6.16 and the assumption that B is an orthonormal basis ofHtherefore assure for all µ P σP pAq “ σP pBq “ Impλq that Kernpµ´Bq “ KernpµÁq.Combining this and (3.184) with Theorem 3.6.17 proves that A “ B. The proof ofProposition 3.6.19 is thus completed.

Issue 3.6.20. LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, let A : DpAq ĎH Ñ H be a closed linear operator, let B Ď DpAq be an orthonormal basis of H, andlet λ : B Ñ K be a function which satisfies for all b P B that Ab “ λbb. Is A then adiagonal linear operator?


3.6.3 Fractional powers of a diagonal linear operator

Definition 3.6.21 (Nonnegative fractional powers of a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P r0,8q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď r0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that for allv P DpArq it holds that

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλÁq,Hpvq›

›

2

Hă 8

,

.

-

(3.187)

andArv “

ÿ

λPσP pAq

λr ¨ PKernpλÁq,Hpvq. (3.188)

Definition 3.6.22 (Negative fractional powers of a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P p´8, 0q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that for allv P DpArq it holds that

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλÁq,Hpvq›

›

2

Hă 8

,

.

-

(3.189)

andArv “

ÿ

λPσP pAq

λr ¨ PKernpλÁq,Hpvq. (3.190)

The next lemma collects a simple property of fractional powers of a diagonallinear operator. It follows immediately from Definition 3.6.21 and Definition 3.6.22.

Lemma 3.6.23 (Diagonality of fractional powers of a diagonal linear operators).Let K P tR,Cu, r P R, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq ĎH Ñ H be a diagonal linear operator with σP pAq Ď r0,8q, and assume that

`

r Pr0,8q or σP pAq Ď p0,8q

˘

. Then Ar is a diagonal linear operator.


3.6.4 Domain Hilbert space associated to a diagonal linearoperator

Lemma 3.6.24. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Thenthe triple

`

DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘

is a K-inner product space.

The proof of Lemma 3.6.24 is clear and therefore omitted. If the point spectrum ofthe diagonal linear operator A in Lemma 3.6.24 in addition satisfies infpσP pAqq ą 0,then the triple

`


is even a K-Hilbert space. This is thesubject of the next lemma.

Lemma 3.6.25 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q andinfpσP pAqq ą 0. Then

(i) it holds that the triple`


is a K-Hilbert space and

(ii) it holds for all v P DpAq that

vH ďAvH

infpσP pAqq. (3.191)

Proof of Lemma 3.6.25*. First of all, note that for all v P DpAq it holds that

Av2H “ÿ

µPσP pAq

›

›µ ¨ PKernpµÁq,Hpvq›

›

2

H“

ÿ

µPσP pAq

|µ|2 ¨›

›PKernpµÁq,Hpvq›

›

2

H

ě

„

infµPσP pAq

|µ|2

»

–

ÿ

µPσP pAq

›

›PKernpµÁq,Hpvq›

›

2

H

fi

fl “

„

infµPσP pAq

µ

2

v2H .

(3.192)

This proves (3.191). Moreover, note that Lemma 3.6.24 ensures that the triple`


is a K-inner product space. It thus remains to provethat the normed K-vector space

`

DpAq, Ap¨qH˘

is complete. For this let pvnqnPN ĎDpAq be a Cauchy sequence in

`

DpAq, Ap¨qH˘

. Inequality (3.191) hence impliesthat pvnqnPN is a Cauchy sequence in pH, ¨Hq too. This and the fact that pH, ¨Hq iscomplete shows that there exists a vector v P H such that lim supnÑ8 vn ´ vH “ 0.


Next note that for all n P N it holds that

8 ą lim infmÑ8

Apvn ´ vmq2H

“ lim infmÑ8

»

–

ÿ

µPσP pAq

|µ|2›

›PKernpµÁq,Hpvn ´ vmq›

›

2

H

fi

fl

ěÿ

µPσP pAq

|µ|2”

lim infmÑ8

›

›PKernpµÁq,Hpvn ´ vmq›

›

2

H

ı

“ÿ

µPσP pAq

|µ|2„

›

›

›PKernpµÁq,H

´

vn ´ limmÑ8

vm

¯›

›

›

2

H

“ÿ

µPσP pAq

|µ|2”

›

›PKernpµÁq,Hpvn ´ vq›

›

2

H

ı

.

(3.193)

Combining this and the fact that lim supnÑ8 lim infmÑ8 Apvn ´ vmq2H “ 0 ă 8

with the fact that @n P N : vn P DpAq shows that v P DpAq and

lim supnÑ8

Apvn ´ vqH “ 0. (3.194)


Exercise 3.6.26 (*). Give an example of an R-Hilbert space pH, 〈¨, ¨〉H , ¨Hq anda diagonal linear operator A : DpAq Ď H Ñ H such that σP pAq Ď p0,8q and suchthat the triple

`


is not an R-Hilbert space.

3.6.5 Interpolation spaces associated to a diagonal linear op-erator

Theorem 3.6.27 (Completion*). Let pE, dEq be a metric space. Then there exists

a complete metric space pF, dF q such that E Ď F , EF“ F , and dF |EÊ “ dE.

Theorem 3.6.27 can be proved by considering the set of equivalence classes ofCauchy sequences in E. The detailed proof of Theorem 3.6.27 is well known andtherefore omitted.

Definition 3.6.28 (Completion*). Let pE, dEq be a metric space and let pF, dF q be

a complete metric space such that E Ď F , EF“ F , and dF |EÊ “ dE. Then and

only then we say that pF, dF q is a completion of pE, dEq.


We now introduce the concept of a family of interpolation spaces associated to adiagonal linear operator.

Theorem 3.6.29 (Interpolation spaces associated to a diagonal linear operator*).LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, and let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0. Then there exists anup to isometric isomorphisms unique family pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, of K-Hilbertspaces with the property that

(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs

,

(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and

(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .

Proof of Theorem 3.6.29. Let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P r0,8q, be theK-Hilbert spaceswith the property that for all r P r0,8q it holds that

pHr, 〈¨, ¨〉Hr , ¨Hrq “ pDpArq, 〈Arp¨q, Arp¨q〉H , A

rp¨qHq. (3.195)

Note that Lemma 3.6.25 ensures that such K-Hilbert spaces do indeed exist. In thenext step let H8 be the set given by H8 “ XrPp0,8qHr. Then we observe that H8is a K-vector space. Furthermore, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P p´8, 0q, be K-Hilbertspaces with the property that for all r P p´8, 0q it holds that

Hr “

H Z

#

pH,ϕq P tHu ˆ LinpH8,Kq :

«

supvPH8zt0u

|ϕpvq|

vH´ră 8 “ sup

vPH8zt0u

|ϕpvq|

vH

ff+

,

(3.196)

with the property that for all r P p´8, 0q, λ P K, v, w P H it holds that

vHr “ supuPH8zt0u

|〈v, u〉H |uH´r

, v `Hr w “ v ` w, λ ¨Hr v “ λ ¨ v, (3.197)

with the property that for all r P p´8, 0q, λ P K, ϕ P LinpH8,Kq with supvPH8zt0u|ϕpvq|vH´r

ă 8 it holds that

pH,ϕqHr “ supuPH8zt0u

|ϕpuq|

uH´r, λ ¨Hr pH,ϕq “ pH, λ ¨ ϕq, (3.198)


and with the property that for all r P p´8, 0q, v P H, ϕ, ψ P LinpH8,Kq with

supvPH8zt0u|ϕpvq|`|ψpvq|vH´r

ă 8 it holds that

pH,ϕq `Hr pH,ψq

“

#

w : rDw P H : @u P H8 : 〈w, u〉H “ ϕpuq ` ψpuqs

pH,ϕ` ψq : else

(3.199)

and

pH,ϕq `Hr v “ v `Hr pH,ϕq “´

H,“

H8 Q u ÞÑ 〈v, u〉H ` ϕpuq P K‰

¯

. (3.200)


Definition 3.6.30 (Interpolation spaces associated to a diagonal linear operator*).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H bea symmetric diagonal linear operator with infpσP pAqq ą 0, and let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be K-Hilbert spaces with the property that

(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs

,

(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and

(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .

Then and only then we say that pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, is a family of interpolationspaces associated to A.

3.7 The Bochner integral

3.7.1 Existence and uniqueness of the Bochner integral

Theorem 3.7.1 (Bochner integral*). Let pΩ,F , µq be a finite measure space, letK P tR,Cu, and let pV, ¨V q be a K-Banach space. Then

(i) there exists a unique continuous K-linear function I : L1pµ; ¨V q Ñ V with theproperty that for all F/BpV q-simple f : Ω Ñ V it holds that

Ipfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v (3.201)

(ii) and it holds for all f P L1pµ; ¨V q that IpfqV ď fL1pµ;¨V q.

3.7. THE BOCHNER INTEGRAL 97

Proof of Theorem 3.7.1*. Throughout this proof let S Ď L1pµ; ¨V q be the set of allF/BpV q-simple functions and let J : S Ñ V be the mapping with the property thatfor all f P S it holds that

Jpfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.202)

Next observe that the triangle inequality proves that for all f P S it holds that

JpfqV ďř

vPfpΩq

µpf´1pvqq ¨ vV “ fL1pµ;¨V q. (3.203)

This, the fact that J is linear, and Lemma 2.4.27 imply that J is uniformly continu-ous. In addition, we note that Item (iv) in Theorem 2.3.10 and Lebesgue’s theoremof dominated convergence ensure that

SL1pµ;¨V q

“ L1pµ; ¨V q. (3.204)

The assumption that V is complete hence allows us to apply Proposition 2.4.22to obtain that there exists a unique I P CpL1pµ; ¨V q, V q with the property thatI|S “ J . This proves (i). In addition, observe that (i), (3.203), and (3.204) establish(ii). The proof of Theorem 3.7.1 is thus completed.

3.7.2 Definition of the Bochner integral

Definition 3.7.2 (*). Let pΩ,F , µq be a finite measure space, let K P tR,Cu, andlet pV, ¨V q be a K-Banach space. Then we denote by

ş

Ωp¨q dµ : L1pµ; ¨V q Ñ V the

continuous K-linear function with the property that for all F/BpV q-simple f : Ω Ñ Vit holds that

ż

Ω

f dµ “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.205)

Corollary 3.7.3 (Triangle inequality for the Bochner integral*). Let pΩ,F , µq bea finite measure space, let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letf P L1pµ; ¨V q. Then

›

›

›

›

ż

Ω

f dµ

›

›

›

›

V

ď

ż

Ω

fV dµ. (3.206)

Corollary 3.7.3 is an immediate consequence of Theorem 3.7.1.


Chapter 4

Semigroups of bounded linearoperators

In this chapter we follow with some minor changes the presentations in Pazy [17].

4.1 Definition of a semigroup of bounded linear

operators

Definition 4.1.1 (Semigroups of bounded linear operators*). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let S : r0,8q Ñ LpV q be a mapping withthe property that for all t1, t2 P r0,8q it holds that

S0 “ IdV and St1St2 “ St1`t2looooooomooooooon

semigroup property

. (4.1)

Then and only then we say that S is a semigroup (we say that S is a semigroup ofbounded linear operators on V ).

4.2 Types of semigroups

Definition 4.2.1 (Contraction semigroups*). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we say thatS is contractive if and only if it holds that

suptPr0,8q

StLpV q ď 1. (4.2)

99

100 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS

Definition 4.2.2 (Strongly continuous semigroups*). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we saythat S is strongly continuous if and only if it holds for every v P V that the function

r0,8q Q t ÞÑ Stv P V (4.3)

is continuous.

Definition 4.2.3 (Uniformly continuous semigroups*). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then we saythat S is uniformly continuous if and only if the function

r0,8q Q t ÞÑ St P LpV q (4.4)

is continuous.

Example 4.2.4 (Matrix exponential*). Let d P N and let A P Rdˆd be an arbitraryd ˆ d-matrix. Then the function r0,8q Q t ÞÑ eAt P Rdˆd is a uniformly continuoussemigroup.

Clearly, it holds that every uniformly continuous semigroup is also strongly con-tinuous. However, not every strongly continuous semigroup is uniformly continuoustoo. This is the subject of the next exercise.

Exercise 4.2.5. Give an example of an R-Banach space pV, ¨V q and a stronglycontinuous semigroup S : r0,8q Ñ LpV q so that S is not a uniformly continuoussemigroup. Prove that your function S does indeed fulfill the desired properties.

4.3 The generator of a semigroup

Definition 4.3.1 (Generator*). Let K P tR,Cu, let pV, ¨V q be a normed K-vectorspace, and let S : r0,8q Ñ LpV q be a semigroup. Then we denote by GS : DpGSq ĎV Ñ V the function with the property that

DpGSq “"

v P V :

ˆ„

Stv ´ v

t

converges as p0,8q Q tŒ 0

*

(4.5)

and with the property that for all v P DpGSq it holds that

GSv “ limtŒ0

„

Stv ´ v

t

(4.6)

and we call GS the generator of S (we call GS the infinitesmal generator of S).

4.4. GLOBAL A PRIORI BOUNDS FOR SEMIGROUPS 101

In the next notion we label all linear operators that are generators of stronglycontinuous semigroups.

Definition 4.3.2 (Generator of a strongly continuous semigroup*). Let K P tR,Cu,let pV, ¨V q be a K-Banach space, and let A : DpAq Ď V Ñ V be a linear operatorwith the property that there exists a strongly continuous semigroup S : r0,8q Ñ LpV qsuch that

GS “ A. (4.7)

Then and only then we say that A is a generator of a strongly continuous semigroup.

We complete this section with a simple exercise which aims to illustrate and relatethe different concepts introduced above.

Exercise 4.3.3 (*). Let K P tR,Cu, let pV, ¨V q be a normed K-vector space with#V pV q ą 1, and let S : r0,8q Ñ LpV q be the function with the property that for allt P r0,8q it holds that

St “

#

IdV : t “ 0

0 : t ą 0. (4.8)

(i) Is S a semigroup? Prove that your answer is correct.

(ii) Is S a strongly continuous semigroup? Prove that your answer is correct.

(iii) Is S a uniformly continuous semigroup? Prove that your answer is correct.

(iv) Is S a contractive semigroup? Prove that your answer is correct.

(v) Specify DpGSq and GS.

4.4 Global a priori bounds for semigroups

In the next result, Proposition 4.4.1, we present a global a priori bound for semigroupsof bounded linear operators.

Proposition 4.4.1 (Global a priori bound*). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then it holds forall t P r0,8q, ε P p0,8q that

supsPr0,ts SsLpV q ď“

supsPr0,εs SsLpV q‰

¨ et“

lnpSε1εLpV qq

‰`

. (4.9)


Proof of Proposition 4.4.1*. Note that for all t P r0,8q, ε P p0,8q, n P N0 X ptε ´

1, tεs it holds that

StLpV q “›

›Snε`ptńεq›

›

LpV q“›

›SnεSptńεq›

›

LpV qď SnεLpV q

›

›Sptńεq›

›

LpV q

“ rSεsnLpV q

›

›Sptńεq›

›

LpV qď Sε

nLpV q

›

›Sptńεq›

›

LpV q

ď“


SεnLpV q ď

“

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰n

ď“

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰tε

““


max!

e0, exp´

t ln´

Sε1εLpV q

¯¯)

ď“


exp´

tmax

0, ln`

Sε1εLpV q

˘(

¯

.

(4.10)

This completes the proof of Proposition 4.4.1.

4.5 Strongly continuous semigroups

4.5.1 A priori bounds for strongly continuous semigroups

In Corollary 4.5.5 below we present a global priori bound for strongly continuoussemigroups. The proof of Corollary 4.5.5 uses the local a priori bound in Lemma 4.5.4below. The proof of Lemma 4.5.4, in turn, exploits the uniform boundedness princi-ple. This is the subject of the next result.

4.5.1.1 The Baire category theorem on complete metric spaces

Lemma 4.5.1 (A set contains an open ball). Let pE, dEq be a metric space andlet A Ď E. Then Ac ‰ E if and only if there exist ε P p0,8q, x P E such thatty P E : dEpx, yq ă εu Ď A.

Proof of Lemma 4.5.1. Observe that

Ac “ E

ô Ac is dense in E

ô @x P E : @ ε P p0,8q : D y P Ac : dEpx, yq ă ε

ô @x P E : @ ε P p0,8q : Ac X ty P E : dEpx, yq ă εu ‰ H.

(4.11)

4.5. STRONGLY CONTINUOUS SEMIGROUPS 103

This implies that

Ac ‰ E

ô Dx P E : D ε P p0,8q : Ac X ty P E : dEpx, yq ă εu “ H

ô Dx P E : D ε P p0,8q : ty P E : dEpx, yq ă εu Ď A.

(4.12)


Theorem 4.5.2 (Baire category theorem for complete metric spaces). Let pE, dEqbe a complete metric space and let An Ď E, n P N, be a sequence of closed subsetsof E with the property that rYnPNAns

c‰ E. Then there exists an N P N such that

rAN sc‰ E.

4.5.1.2 The uniform boundedness principle

Theorem 4.5.3 (Uniform boundedness principle*). Let K P tR,Cu, let pU, Üq bea K-Banach space, let pV, ¨V q be a normed K-vector space, and let A Ď LpU, V q bea non-empty set with the property that for all u P U it holds that

supAPA

AuV ă 8. (4.13)

Then

supAPA

ALpU,V q ă 8. (4.14)

4.5.1.3 Local a priori bounds

Lemma 4.5.4 (Local a priori bound*). Let K P tR,Cu, let pV, ¨V q be a K-Banachspace, and let S : r0,8q Ñ LpV q be a semigroup which satisfies for all v P V thatlim suptŒ0 Stv ´ vV “ 0. Then

lim suptŒ0

StLpV q “ limtŒ0

supsPr0,ts

SsLpV q ă 8. (4.15)

Proof of Lemma 4.5.4*. We prove Lemma 4.5.4 by a contradiction. More specifi-cally, we assume in the following that

limtŒ0

supsPr0,ts

SsLpV q “ 8. (4.16)


This and the fact that S0LpV q “ 1 ă 8 imply that for all t P p0,8q it holds that

supsPp0,ts

SsLpV q “ 8. (4.17)

Hence, there exists a strictly decreasing sequence tn P p0,8q, n P N, with limnÑ8 tn “0 and with the property that for all n P N it holds that

StnLpV q ě n. (4.18)

This ensures thatsupnPN

StnLpV q “ 8. (4.19)

Theorem 4.5.3 hence implies that there exists a vector v P V such that

supnPN

StnvV “ 8. (4.20)

Combining this and the fact that @n P N : StnvV ă 8 implies that

lim supnÑ8

StnvV “ 8. (4.21)

This and the assumption that @ v P V : limtŒ0 Stv “ v show that

8 ą vV “›

›

›limnÑ8

rStnvs›

›

›

V“ lim

nÑ8StnvV “ lim sup

nÑ8StnvV “ 8. (4.22)

This contradiction completes the proof of Lemma 4.5.4.

4.5.1.4 Global a priori bounds

The next result, Corollary 4.5.5, proves a stronger version of Lemma 4.5.4. Observethat Lemma 4.5.4 and Corollary 4.5.5 apply to strongly continuous semigroups onBanach spaces.

Corollary 4.5.5 (Global a priori bound*). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup which satisfies for all v P Vthat limtŒ0 Stv “ v. Then it holds for all t P r0,8q, ε P p0,8q that

supsPr0,ts SsLpV q ď“


¨ et“

lnpSε1εLpV qq

‰`

ă 8. (4.23)

Corollary 4.5.5 is an immediate consequence of Proposition 4.4.1 and Lemma 4.5.4above.


4.5.2 Existence of solutions of linear ordinary differentialequations in Banach spaces

Lemma 4.5.6 (Invariance of the domain of the generator*). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup.Then it holds for all t P r0,8q, v P DpGSq that

St`

DpGSq˘

Ď DpGSq and GSStv “ StGSv. (4.24)

Proof of Lemma 4.5.6*. Observe that for all t P r0,8q, v P DpGSq it holds that

limsŒ0

„

Ss rStvs ´ rStvs

s

“ limsŒ0

„

St

„

Ssv ´ v

s

“ St

„

limsŒ0

„

Ssv ´ v

s

“ StGSv.

(4.25)This completes the proof of Lemma 4.5.6.

Lemma 4.5.7 (*). LetK P tR,Cu, let pV, ¨V q be aK-Banach space, let S : r0,8q ÑLpV q be a strongly continuous semigroup, and let v P DpGSq. Then

(i) it holds that the function r0,8q Q t ÞÑ Stv P V is continuously differentiableand

(ii) it holds for all t P r0,8q that

ddtrStvs “ GSStv “ StGSv. (4.26)

Proof of Lemma 4.5.7*. Observe that for all s, t P r0,8q with s ‰ t it holds that

›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

“

›

›

›

›

Smints,tu

„

Ss´mints,tuv ´ St´mints,tuv

s´ t

´ GSStv›

›

›

›

V

ď

›

›

›

›

Sminps,tq

„

Smaxts,tu´mints,tuv ´ v

maxts, tu ´mints, tu´ GSv

›

›

›

›

V

`›

›

“

Smints,tu ´ St‰

GSv›

›

V

ď›

›Sminps,tq

›

›

LpV q

›

›

›

›



›

›

›

›

V

`›

›

“

Smints,tu ´ St‰

GSv›

›

V.

(4.27)


Corollary 4.5.5 and the fact that S is strongly continuous hence imply that for allt P r0,8q it holds that

lim supr0,8qzttuQsÑt

›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

ď

«

supsPr0,t`1s

SsLpV q

ff«

lim supr0,8qzttuQsÑt

›

›

›

›



›

›

›

›

V

ff

` lim supr0,8qzttuQsÑt

›

›

“

Smints,tu ´ St‰

GSv›

›

V“ 0.

(4.28)


4.5.3 Pointwise convergence in the space of bounded linearoperators

Lemma 4.5.8 (A characterization of pointwise convergence in the space of boundedlinear operators). LetK P tR,Cu, let pV, ¨V q be aK-Banach space, and let pSnqnPN0 Ď

LpV q. Then @ v P V : limnÑ8 Snv ´ S0vV “ 0 if and only if for all compact setsK Ď V it holds that limnÑ8 supvPK Snv ´ S0vV “ 0.

Proof of Lemma 4.5.8. The proof of the “ð” direction in the statement of Lemma 4.5.8is clear. It thus remains to prove the “ñ” direction in the statement of Lemma 4.5.8.To this end we assume that for all v P V it holds that limnÑ8 Snv “ S0v and weassume that there exists a compact set K Ď V such that

lim supnÑ8

supvPK

Snv ´ S0vV ą 0. (4.29)

In the next step we note that there exists a sequence pvnqnPN Ď K such that for alln P N it holds that

Snvn ´ S0vnV “ supvPK

Snv ´ S0vV . (4.30)

The compactness of K ensures that there exist a w P K and a strictly increasingsequence pnkqkPN Ď N such that limkÑ8 vnk “ w. By assumption it holds that


limkÑ8 Snkw “ S0w. This and Theorem 4.5.3 imply that

0 “ lim supkÑ8

Snkw ´ S0wV

“ lim supkÑ8

Snkpw ´ vnkq ` pSnk ´ S0q vnk ` S0pvnk ´ wqV

ě lim supkÑ8

pSnk ´ S0q vnkV ´ lim supkÑ8

Snkpw ´ vnkqV ´ limkÑ8

S0 pvnk ´ wqV

ě lim supkÑ8

supvPK

pSnk ´ S0q vV ´

„

supkPN

SnkLpV q

lim supkÑ8

w ´ vnkV

“ lim supkÑ8

supvPK

pSnk ´ S0q vV ą 0.

(4.31)

This condradiction completes the proof of Lemma 4.5.8.

4.5.4 Domains of generators of strongly continuous semi-groups

In this subsection we prove that the generator of a strongly continuous semigroup isdensily defined ; see Corollary 4.5.10 below. In the proof of Corollary 4.5.10 we usethe following result, Lemma 4.5.9. Lemma 4.5.9 and its proof can, e.g., be found asTheorem 2.4 (b) in Pazy [17] and Corollary 4.5.10 and its proof can, e.g., be foundas Corollary 2.5 in Pazy [17].

Lemma 4.5.9 (Fundamental theorem of calculus for strongly continuous semi-groups). LetK P tR,Cu, t P r0,8q, let pV, ¨V q be aK-Banach space, let S : r0,8q Ñ

LpV q be a strongly continuous semigroup, and let v P V . Then it holds thatşt

0Ssv ds P

DpGSq and it holds that

GSˆż t

0

Ssv ds

˙

“ Stv ´ v. (4.32)

Proof of Lemma 4.5.9. Throughout this proof we assume w.l.o.g. that t P p0,8q.Then we observe that for all u P p0, tq it holds that

rSu ´ IdV s

u

„ż t

0

Ssv ds

“1

u

ż t

0

rSu`sv ´ Ssvs ds “1

u

ż tù

t

Ssv ds´1

u

ż u

0

Ssv ds.

(4.33)


Continuity of the function r0,8q Q s ÞÑ Ssv P V hence proves thatşt

0Ssv ds P DpGSq

and that

GSˆż t

0

Ssv ds

˙

“ limuŒ0

„

rSu ´ IdV s

u

„ż t

0

Ssv ds

“ Stv ´ S0v “ Stv ´ v. (4.34)


We are now ready to prove that the generator of a strongly continuous semigroupis densily defined.

Corollary 4.5.10. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letS : r0,8q Ñ LpV q be a strongly continuous semigroup. Then DpGSq is dense in V .

Proof of Corollary 4.5.10. Let v P V be arbitrary. The assumption that S is astrongly continuous semigroup together with the fundamental theorem of calculusensures that

limtŒ0

ˆ

1

t

ż t

0

Ssv ds

˙

“ v. (4.35)

In addition, Lemma 4.5.9 proves that for all t P p0,8q it holds that 1t

şt

0Ssv ds P

DpGSq. This and (4.35) imply that v P DpGSq. The proof of Corollary 4.5.10 is thuscompleted.

4.5.5 Generators of strongly continuous semigroups

In this section we show that a strongly continuous semigroup is uniquely deter-mined by its generator; see Proposition 4.5.12 below. In Proposition 4.5.12 we usethe assumption that the graph of one mapping is a subset of the graph of anothermapping. To getter a better understanding for this assumption, we first note thefollowing remark.

Remark 4.5.11 (*). Let A1, A2, B be sets and let f1 : A1 Ñ B and f2 : A2 Ñ B bemappings. Then it holds that Graphpf1q Ď Graphpf2q if and only if (A1 Ď A2 andf2|A1 “ f1).

We are now ready to show that a strongly continuous semigroup is uniquelydetermined by its generator.

Proposition 4.5.12 (The generator determines the semigroup*). Let K P tR,Cu,let pV, ¨V q be a K-Banach space, and let S, S : r0,8q Ñ LpV q be strongly continuoussemigroups with GraphpGSq Ď GraphpGSq. Then it holds that S “ S and GS “ GS.


Proof of Proposition 4.5.12. Let v P DpGSq Ď DpGSq, t P p0,8q and let η : r0, ts Ñ Vbe the function with the property that for all s P r0, ts it holds that

ηpsq “ St´s Ss v. (4.36)

Then it holds for all s P r0, ts, u P r0, ts with s ‰ u that

›

›

›

›

ηpuq ´ ηpsq

u´ s

›

›

›

›

V

“

›

›

›

›

›

Stú Su v ´ St´s Ss v

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

Stú

„

Su v ´ Ss v

u´ s

`

“

Stú ´ St´s‰

Ssv

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

St´s

„

Su v ´ Ss v

u´ s

`

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

ď

›

›

›

›

›

St´s

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

`

›

›

›

›

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

›

›

›

›

V

.

(4.37)

This implies that for all s P r0, ts, punqnPN Ď r0, tsztsu with limnÑ8 un “ s and alln P N it holds that

›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď

›

›

›

›

›

St´s

„

Sunv ´ Ss v

un ´ s

´

“

Stún ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` sup!

›

›Stúnw ´ St´sw›

›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(4.38)

Lemma 4.5.7 and Lemma 4.5.6 prove that for all s P r0, ts it holds that

limuÑs

«

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

ff

“ GSSt´sSsv “ St´sGSSsv “ St´sGSSsv (4.39)

and

limuÑs

„

Suv ´ Ss v

u´ s

“ GSSsv. (4.40)


Putting (4.39)–(4.40) into (4.38) proves that for all s P r0, ts and all punqnPN Ď

r0, tsztsu with limnÑ8 un “ s it holds that

limnÑ8

›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď limnÑ8

›

›

›

›

›

St´s

„

Sunv ´ Ss v

un ´ s

´

“

Stún ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“

›

›

›St´sGSSsv ´ St´sGSSsv

›

›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“ lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(4.41)

This together with Lemma 4.5.7 and Lemma 4.5.8 proves that η is differentiable andthat for all s P r0, ts it holds that η1psq “ 0. This implies that

Stv “ ηp0q “ ηptq “ Stv. (4.42)

As v P DpGSq was arbitrary, we obtain that St|DpGSq “ St|DpGSq. Corollary 4.5.10

hence proves that St “ St. This completes the proof of Proposition 4.5.12.

Lemma 4.5.13 (Closedness of generators of strongly continuous semigroups). LetK P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be astrongly continuous semigroup. Then GS is a closed linear operator.

Proof of Lemma 4.5.13. Throughout this proof let x, y P V and let pvnqnPN Ď DpGSqbe a sequence which satisfies lim supnÑ8 x´vnV “ lim supnÑ8 y´GSvnV “ 0. Itis clear that GS is a linear operator. Next note that Lemma 4.5.7 and the fundamentaltheorem of calculus show for all t P r0,8q, n P N that

Stvn ´ vn “

ż t

0

SsGSvn ds. (4.43)

Moreover, Corollary 4.5.5 implies for all t P r0,8q that

lim supnÑ8

›

›

›

›

ż t

0

Ssy ds´

ż t

0

SsGSvn ds›

›

›

›

V

ď lim supnÑ8

ż t

0

Ssy ´ SsGSvnV ds

ď t ¨“

supsPr0,ts SsLpV q‰“

lim supnÑ8 y ´ GSvnV‰

“ 0.

(4.44)


This and (4.43) prove for all t P r0,8q that

Stx´ x “

ż t

0

Ssy ds. (4.45)

Again the fundamental theorem of calculus hence shows that

lim suptŒ0

›

›

›

›

Stx´ x

t´ y

›

›

›

›

V

“ lim suptŒ0

›

›

›

›

ˆ

1

t

ż t

0

Ssy ds

˙

´ y

›

›

›

›

V

“ 0. (4.46)

This ensures that x P DpGSq and GSx “ y. The proof of Lemma 4.5.13 is thuscompleted.

4.5.6 A generalization of matrix exponentials to infinite di-mensions

Proposition 4.5.12 and Definition 4.3.2 ensure that the next definition, Definition 4.5.14,makes sense.

Definition 4.5.14 (Generalized matrix exponential*). Let K P tR,Cu, let pV, ¨V qbe a K-Banach space, and let A : DpAq Ď V Ñ V be a generator of a stronglycontinuous semigroup. Then we denote by eAt P LpV q, t P r0,8q, the linear opera-tors with the property that for all t P r0,8q and all strongly continuous semigroupsS : r0,8q Ñ LpV q with GS “ A it holds that

etA “ St. (4.47)

4.5.7 A characterization of strongly continuous semigroups

Lemma 4.5.15 (Characterization of strongly continuous semigroups*). Let pV, ¨V qbe a Banach space. A semigroup S : r0,8q Ñ LpV q is strongly continuous if and onlyif for all v P V it holds that lim suptŒ0 Stv ´ vV “ 0.

Proof of Lemma 4.5.15*. A strongly continuous semigroup S : r0,8q Ñ LpV q clearlysatisfies that for all v P V it holds that limtŒ0 Stv “ v. In the following we thusassume that S : r0,8q Ñ LpV q is a semigroup which fulfills for all v P V that


limtŒ0 Stv “ v. Corollary 4.5.5 hence implies that for all t P r0,8q it holds that

lim supsÑt

Ssv ´ StvV “ lim supsÑt

›

›Sminps,tq

`

S|t´s|v ´ v˘›

›

V

ď lim supsÑt

”

›

›Sminps,tq

›

›

LpV q

›

›S|t´s|v ´ v›

›

V

ı

ď

«

supuPr0,t`1s

SuLpV q

ff

„

lim supsÑt

›

›S|t´s|v ´ v›

›

V

“ 0.

(4.48)


4.6 Uniformly continuous semigroups

Lemma 4.6.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup with the property that

limtŒ0St ´ S0LpV q “ 0. (4.49)

Then it holds for all t P r0,8q that supsPr0,ts SsLpV q ă 8.

Proof of Lemma 4.6.1. The assumption limtŒ0 St ´ S0LpV q “ 0 ensures that there

exists a real number ε P p0,8q such that

supsPr0,εs

SsLpV q ă 8. (4.50)

Combining this with Proposition 4.4.1 completes the proof of Lemma 4.6.1.

Lemma 4.6.2. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup. Then S is uniformly continuous if and only iflimtŒ0 St ´ S0LpV q “ 0.

Proof of Lemma 4.6.2. Clearly, it holds that if S is uniformly continuous, then itholds that limtŒ0 St ´ S0LpV q “ 0. It thus remains to prove that the condition

limtŒ0 St ´ S0LpV q “ 0 ensures that S is uniformly continuous. We thus assume inthe following that

limtŒ0St ´ S0LpV q “ 0. (4.51)

Lemma 4.6.1 hence implies that for all t P r0,8q it holds that

supsPr0,ts

SsLpV q ă 8. (4.52)

4.6. UNIFORMLY CONTINUOUS SEMIGROUPS 113

This and (4.51) show that for all t P r0,8q it holds that

limsÑtSs ´ StLpV q “ lim

sÑt

›

›Sminps,tq

“

Srmaxps,tq´minps,tqs ´ S0

‰›

›

LpV q

ď

”

limsÑt

›

›Srmaxps,tq´minps,tqs ´ S0

›

›

LpV q

ı

«

supsPr0,t`1s

SsLpV q

ff

“ 0.(4.53)


4.6.1 Matrix exponential in Banach spaces

The next result, Lemma 4.6.3, demonstrates one way how uniformly continuoussemigroup can be constructed.

Lemma 4.6.3 (Matrix exponential in Banach spaces). Let K P tR,Cu, let pV, ¨V qbe a K-Banach space, and let A P LpV q. Then

(i) it holds that A is a generator of a strongly continuous semigroup,

(ii) it holds that peAtqtPr0,8q Ď LpV q is a uniformly continuous semigroup,

(iii) it holds for all t P r0,8q that›

›eAt›

›

LpV qďř8

n“0

›

›

pAtqn

n!

›

›

LpV q“ et ALpV q ă 8, and

(iv) it holds for all t P r0,8q that

eAt “8ÿ

n“0

pAtqn

n!. (4.54)

Proof of Lemma 4.6.3. First, note that

8ÿ

n“0

›

›

›

›

pAtqn

n!

›

›

›

›

LpV q

“

8ÿ

n“0

tn AnLpV qn!

“ etALpV q ă 8. (4.55)

Next let S : r0,8q Ñ LpV q be the function with the property that for all t P r0,8qit holds that

St “8ÿ

n“0

pAtqn

n!. (4.56)


Observe that for all t1, t2 P r0,8q it holds that

St1St2 “

«

8ÿ

n“0

pAt1qn

n!

ff«

8ÿ

n“0

pAt2qn

n!

ff

“

8ÿ

n,m“0

An`m pt1qnpt2q

m

n!m!

“

8ÿ

k“0

ÿ

n,mPN0n`m“k

Ak pt1qnpt2q

m

n!m!“

8ÿ

k“0

Ak

k!

«

kÿ

n“0

k!

n! pk ´ nq!¨ pt1q

n¨ pt2q

pkńq

ff

“

8ÿ

k“0

Ak

k!rt1 ` t2s

k“ St1`t2 .

(4.57)

This shows that S is a semigroup. Moreover, observe that for all t P r0,8q it holdsthat

St ´ S0LpV q “

›

›

›

›

›

8ÿ

n“1

pAtqn

n!

›

›

›

›

›

LpV q

ď t ALpV q

›

›

›

›

›

8ÿ

n“1

pAtqpn´1q

n!

›

›

›

›

›

LpV q

“ t ALpV q

›

›

›

›

›

8ÿ

n“0

pAtqn

pn` 1q!

›

›

›

›

›

LpV q

ď t ALpV q

«

8ÿ

n“0

AtnLpV qpn` 1q!

ff

ď t ALpV q etALpV q .

(4.58)

This proves that S is uniformly continuous. Furthermore, note that for all t P p0,8qit holds that

›

›

›

›

St ´ S0

t´ A

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

pAtqpn´1q

n!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“0

pAtqn

pn` 1q!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

pAtqn

pn` 1q!

ff›

›

›

›

›

LpV q

“ t

›

›

›

›

›

A2

«

8ÿ

n“0

pAtqn

pn` 2q!

ff›

›

›

›

›

LpV q

ď t A2LpV q e

tALpV q .

(4.59)

Therefore, we obtain thatGS “ A. (4.60)

This, in turn, establishes Item (i). Proposition 4.5.12, (4.60), and the fact that Sis a uniformly continuous semigroup hence prove Item (ii) and Item (iv). Next notethat Item (iii) follows from Item (iv) and (4.55). The proof of Lemma 4.6.3 is thuscompleted.


4.6.2 Continuous invertibility of bounded linear operatorsin Banach spaces

Lemma 4.6.4 (Geometric series in Banach spaces and inversion of bounded linearoperators). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A P LpV q bea bounded linear operator with IdV ÁLpV q ă 1. Then it holds that A is bijective,

it holds that A´1 P LpV q, it holds thatř8

n“0 rIdV ÁsnLpV q ă 8, and it holds that

A´1“

8ÿ

n“0

rIdV Ásn . (4.61)

Proof of Lemma 4.6.4. Throughout this proof let Q P LpV q and Sn P LpV q, n P N0,be the bounded linear operators with the property that for all n P N it holds that

Q “ IdV Á and Sn “nÿ

k“0

Qk. (4.62)

Note that the assumption that QLpV q ă 1 ensures that

8ÿ

k“0

›

›Qk›

›

LpV qď

8ÿ

k“0

QkLpV q “1

“

1´ QLpV q‰ ă 8. (4.63)

This implies that Sn, n P N0, is a Cauchy-sequence in LpV q and thus convergence inLpV q. Next we claim that for all n P N0 it holds that

ASn “ IdV ´Qn`1. (4.64)

We show (4.64) by induction on n P N0. For this observe that

AS0 “ A IdV “ A “ IdV ´Q. (4.65)

This proves the base case n “ 0 in (4.64). Next note that if n P N and if ASn´1 “

IdV ´Qn, then it holds that

ASn “ A

«

nÿ

k“0

Qk

ff

“ A

«

IdV `nÿ

k“1

Qk

ff

“ A

«

IdV `Q

«

n´1ÿ

k“0

Qk

ffff

“ A rIdV `QSn´1s “ A`QASn´1 “ A`Q rIdV ´Qns

“ A`Q´Qn`1“ IdV ´Q

n`1.

(4.66)


This proves (4.64). Next note that (4.64) implies that for all n P N0 it holds that

ASn “ SnA “ IdV ´Qn`1. (4.67)

This and the fact that pSnqnPN0 Ď LpV q converges shows that

A”

limnÑ8

Sn

ı

loooomoooon

PLpV q

“

”

limnÑ8

Sn

ı

loooomoooon

PLpV q

A “ IdV . (4.68)

This implies that A is bijective and that A´1 “ limnÑ8 Sn P LpV q. The proof ofLemma 4.6.4 is thus completed.

4.6.3 Generators of uniformly continuous semigroup

Lemma 4.6.5 (The generator of a uniformly continuous semigroup). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then GS P LpV q.

Proof of Lemma 4.6.5. The assumption that S is uniformly continuous implies thatfor all t P r0,8q it holds that

limsŒ0

›

›

›

›

1

s

ż t`s

t

Su du´ St

›

›

›

›

LpV q

“ limsŒ0

›

›

›

›

1

s

ż t`s

t

rSu ´ Sts du

›

›

›

›

LpV q

ď limsŒ0

„

1

s

ż t`s

t

Su ´ StLpV q du

ď limsŒ0

«

supuPrt,t`ss

Su ´ StLpV q

ff

ď StLpV q

«

limsŒ0

«

supuPr0,ss

Su ´ S0LpV q

ffff

“ StLpV q

„

lim supsŒ0

Ss ´ S0LpV q

“ StLpV q

„

limsŒ0

Ss ´ S0LpV q

“ 0.

(4.69)

This implies that there exists a real number ε P p0,8q such that›

›

›

›

1

ε

ż ε

0

Ss ds´ S0

›

›

›

›

LpV q

ă 1. (4.70)

Lemma 4.6.4 hence shows thatşε

0Ss ds P LpV q is bijective with

„ż ε

0

Ss ds

´1

P LpV q. (4.71)


Hence, we obtain that for all t P p0, εq it holds that

St ´ IdVt

“

„

St ´ S0

t

„ż ε

0

Ss ds

„ż ε

0

Ss ds

´1

“

„

şε

0rSt`s ´ Sss ds

t

„ż ε

0

Ss ds

´1

“

«

şt`ε

tSs ds´

şε

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

“

«

şε`t

εSs ds´

şt

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

.

(4.72)

This together with the identity (4.69) shows that

limtŒ0

›

›

›

›

›

St ´ IdVt

´ rSε ´ S0s

„ż ε

0

Ss ds

´1›

›

›

›

›

LpV q

“ 0. (4.73)

This proves that GS P LpV q and that

GS “ rSε ´ S0s

„ż ε

0

Ss ds

´1

. (4.74)


4.6.4 A characterization result for uniformly continuous semi-groups

Theorem 4.6.6 (Characterization of uniformly continuous semigroups). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup.Then the following statements are equivalent:

(i) It holds that S is uniformly continuous.

(ii) It holds that limtŒ0 St ´ S0LpV q “ 0.

(iii) It holds that GS P LpV q.


Proof of Theorem 4.6.6. Lemma 4.6.2 implies that (i) and (ii) are equivalent. More-over, Lemma 4.6.5 ensures that (i) implies (iii). It thus remains to prove that (iii)implies (i). To show this we assume for the rest of this proof that GS P LpV q. Thenlet S : r0,8q Ñ LpV q be the function with the property that for all t P r0,8q it holdsthat

St “8ÿ

n“0

ptGSqn

n!. (4.75)

Observe that Lemma 4.6.3 shows that S is uniformly continuous and that

GS “ GS. (4.76)

In the next step we apply Proposition 4.5.12 to obtain that S “ S. This proves thatS is uniformly continuous. The proof of Theorem 4.6.6 is thus completed.

4.6.5 An a priori bound for uniformly continuous semigroups

Combining Lemma 4.6.3 and Theorem 4.6.6 immediately results in the followingestimate.

Proposition 4.6.7 (A priori bounds for uniformly continuous semigroups). Let K PtR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then it holds for all t P r0,8q that

supsPr0,ts

SsLpV q ď et GSLpV q ă 8. (4.77)

4.7 The Hille-Yosida theorem

Definition 4.7.1 (Resolvent set of a linear operator). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Thenwe denote by ρpAq the set given by

ρpAq “

λ P K :`

λ´ A : DpAq Ñ V is bijective and pλ´ Aq´1P LpV q

˘(

(4.78)

and we call ρpAq the resolvent set of A.

Definition 4.7.2 (Yosida approximations of a linear operator). Let K P tR,Cu,let pV, ¨V q be a normed K-vector space, let A : DpAq Ď V Ñ V be a linear opera-tor, and let A : ρpAq Ñ LpV q be a function. Then we call A the family of Yosidaapproximations of A if and only if it holds for all λ P ρpAq that Aλ “ λ2pλÁq´1´λ.

4.7. THE HILLE-YOSIDA THEOREM 119

Remark 4.7.3. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, letA : DpAq Ď V Ñ V be a linear operator, and let pAλqλPρpAq Ď LpV q be the fam-ily of Yosida approximations of A. Then it holds for all λ P ρpAq that Aλ “Ap1´ 1λ ¨ Aq´1 “ λApλ´ Aq´1.

Lemma 4.7.4 (Scalar shifts of generators of strongly continuous semigroups). LetK P tR,Cu, λ P K, let pV, ¨V q be a K-Banach space, and let A : DpAq Ď V Ñ V bea generator of a strongly continuous semigroup. Then it holds that A ´ λ : DpAq ĎV Ñ V is a generator of a strongly continuous semigroup and it holds for all t P r0,8qthat etpA´λq “ e´λtetA.

Proof of Lemma 4.7.4. Throughout this proof let S : r0,8q Ñ LpV q be the mappingwhich satisfies for all t P r0,8q that St “ e´λtetA. Note that S0 “ IdV and that itholds for all t1, t2 P r0,8q that

St1St2 “ e´λt1e´λt2et1Aet2A “ e´λpt1`t2qept1`t2qA “ St1`t2 . (4.79)

Moreover, it holds for all v P V that the function r0,8q Q t ÞÑ Stv P V is continuous.This and (4.79) show that S is a strongly continuous semigroup. In addition, it holdsfor all v P DpAq that

lim suptŒ0

›

›

›

›

Stv ´ v

t´ pA´ λqv

›

›

›

›

V

“ lim suptŒ0

›

›

1tpe´λtetAv ´ vq ´ pA´ λqv

›

›

V

“ lim suptŒ0

›

›

“

1tpe´λt ´ 1q ` λ

‰

etAv ` 1tpetAv ´ vq ´ Av ` λpv ´ etAvq

›

›

V

ď lim suptŒ0

ˇ

ˇ

1tpe´λt ´ 1q ` λ

ˇ

ˇ lim suptŒ0

etAvV

` lim suptŒ0

›

›

1tpetAv ´ vq ´ Av

›

›

V` |λ| ¨ lim sup

tŒ0v ´ etAvV “ 0.

(4.80)

This implies that DpAq Ď DpGSq. Furthermore, it holds for all v P DpGSq that

lim suptŒ0

›

›

›

›

etAv ´ v

t´ pGS ` λqv

›

›

›

›

V

“ lim suptŒ0

›

›

“

1tp1´ e´λtq ´ λ

‰

etAv ` 1tpe´λtetAv ´ vq ´ GSv ` λpetAv ´ vq

›

›

V

ď lim suptŒ0

ˇ

ˇ

1tp1´ e´λtq ´ λ

ˇ

ˇ lim suptŒ0

etAvV

` lim suptŒ0

›

›

1tpStv ´ vq ´ GSv

›

›

V` |λ| ¨ lim sup

tŒ0etAv ´ vV “ 0.

(4.81)

This proves that DpAq Ě DpGSq. Combining (4.80) and (4.81) we hence obtain thatDpAq “ DpGSq and GS “ A ´ λ. Proposition 4.5.12 thus completes the proof ofLemma 4.7.4.


Lemma 4.7.5. LetK P tR,Cu, let pV, ¨V q be aK-Banach space, and let A : DpAq ĎV Ñ V be a linear operator which satisfies that DpAq is dense in V , p0,8q ĎρpAq, and supλPp0,8q p1´ 1λ ¨ Aq´1LpV q ď 1. Then it holds for all v P V that

lim supλÑ8 p1´ 1λ ¨ Aq´1v ´ vV “ 0.

Proof of Lemma 4.7.5. Note that it holds for all v P DpAq that

lim supλÑ8

p1´ 1λ ¨ Aq´1v ´ vV “ lim supλÑ8

1λAp1´ 1λ ¨ Aq´1vV

“ lim supλÑ8

1λp1´ 1λ ¨ Aq´1AvV ď lim sup

λÑ8

1λAvV “ 0.

(4.82)

This implies for all x P V , pvnqnPN Ď DpAq with lim supnÑ8 x´ vnV “ 0 that

lim supλÑ8

p1´ 1λ ¨ Aq´1x´ xV

“ lim supnÑ8

lim supλÑ8

p1´ 1λ ¨ Aq´1px´ vnq ` p1´ 1λ ¨ Aq´1vn ´ vn ` vn ´ xV

ď 2 ¨ lim supnÑ8

x´ vn ` lim supnÑ8

lim supλÑ8

p1´ 1λ ¨ Aq´1vn ´ vnV “ 0. (4.83)


Corollary 4.7.6 (Approximation property of Yosida approximations). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, let A : DpAq Ď V Ñ V be a linear operatorsuch that DpAq is dense in V , p0,8q Ď ρpAq, and supλPp0,8q p1´ 1λ ¨ Aq´1LpV q ď 1,

and let pAλqλPρpAq Ď LpV q be the family of Yosida approximations of A. Then it holdsfor all v P DpAq that lim supλÑ8 Aλv ´ AvV “ 0.

Proof of Corollary 4.7.6. Lemma 4.7.5 and Remark 4.7.3 prove for all v P DpAq that

lim supλÑ8

Aλv ´ AvV “ lim supλÑ8

Ap1´ 1λ ¨ Aq´1v ´ AvV

“ lim supλÑ8

p1´ 1λ ¨ Aq´1Av ´ AvV “ 0.(4.84)


Lemma 4.7.7. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, let A : DpAq ĎV Ñ V be a linear operator such that p0,8q Ď ρpAq and supλPp0,8q p1´ 1λ ¨ Aq´1LpV qď 1, and let pAλqλPρpAq Ď LpV q be the family of Yosida approximations of A. Then itholds for all λ, µ P p0,8q, t P r0,8q, v P V that pesAλqsPr0,8q Ď LpV q is a uniformlycontinuous contraction semigroup and

etAλv ´ etAµvV ď t Aλv ´ AµvV . (4.85)


Proof of Lemma 4.7.7. The fact that pAλqλPp0,8q Ď LpV q and Lemma 4.6.3 show forall λ P p0,8q that petAλqtPr0,8q is a uniformly continuous semigroup. In addition,Lemma 4.7.4 and Item (iii) in Lemma 4.6.3 ensure for all λ P p0,8q, t P r0,8q that

etAλLpV q “ e´λtetrλ2pλÁq´1s

LpV q ď e´λtet λ2pλÁq´1LpV q

“ e´λteλt p1´1λÄq´1LpV q ď e´λteλt “ 1.

(4.86)

This proves that petAλqtPr0,8q is a contraction semigroup. Next note that Lemma 4.5.7and the product rule ensure for all λ, µ P p0,8q, t P r0,8q, v P V that the func-tion r0, 1s Q s ÞÑ etsAλetp1´sqAµv P V is continuously differentiable. The productrule, Lemma 4.5.7, the fact that @λ, µ P p0,8q : AλAµ “ AµAλ, and Item (iv) inLemma 4.6.3 hence show for all λ, µ P p0,8q, t P r0,8q, s P r0, 1s, v P V that

ddsretsAλetp1´sqAµvs “ etsAλAλe

tp1´sqAµv ¨ t´ etsAλetp1´sqAµAµv ¨ t

“ t etsAλetp1´sqAµpAλ ´ Aµqv.(4.87)

This and (4.86) imply for all λ, µ P p0,8q, t P r0,8q, v P V that

etAλv ´ etAµvV “

›

›

›

›

ż 1

0

ddsretsAλetp1´sqAµvs ds

›

›

›

›

V

ď t

ż 1

0

etsAλetp1´sqAµpAλ ´ AµqvV ds ď t Aλv ´ AµvV .

(4.88)


Theorem 4.7.8 (Hille-Yosida). Let K P tR,Cu, let pV, ¨V q be a K-Banach space,and let A : DpAq Ď V Ñ V be a linear operator. Then it holds that A is a generatorof a strongly continuous contraction semigroup if and only if

`

it holds that DpAq isdense in V , it holds that A is a closed linear operator, it holds that p0,8q Ď ρpAq,and it holds that suphPp0,8q p1´ hAq

´1LpV q ď 1˘

.

Proof of Theorem 4.7.8. We first prove the “ñ” direction in the statement of The-orem 4.7.8. To this end assume that A is a generator of a strongly continuouscontraction semigroup. Corollary 4.5.10 and Lemma 4.5.13 prove that DpAq is densein V and that A is a closed linear operator. Furthermore, observe that Lemma 4.7.4shows for all λ P p0,8q, t1 P r0,8q, t2 P rt1,8q, v P V that

›

›

›

›

ż t2

0

espA´λqv ds´

ż t1

0

espA´λqv ds

›

›

›

›

V

“

›

›

›

›

ż t2

t1

e´λsesAv ds

›

›

›

›

V

ď vV

ż t2

t1

e´λs ds “ 1λpe´λt1 ´ e´λt2qvV ď

1λe´λt1vV .

(4.89)


This proves for all λ P p0,8q, v P V that`şn

0espA´λqv ds

˘

nPNis a Cauchy sequence in

pV, ¨V q. Next we introduce some additional notation. Let Rλ P LpV q, λ P p0,8q,be the bounded linear operators which satisfy for all λ P p0,8q, v P V that

Rλv “ limnÑ8

ż n

0

espA´λqv ds. (4.90)

Note that Lemma 4.7.4 implies for all λ P p0,8q, v P V that A´ λ is a generator ofa strongly continuous contraction semigroup and

lim supnÑ8

enpA´λqvV ď lim supnÑ8

e´λnvV “ 0. (4.91)

Lemma 4.5.9 hence ensures for all λ P p0,8q, v P V that`şn

0espA´λqv ds

˘

nPNĎ

DpA´ λq “ DpAq “ Dpλ´ Aq and

lim supnÑ8

›

›

›

›

pλ´ Aq

ˆż n

0

espA´λqv ds

˙

´ v

›

›

›

›

V

“ lim supnÑ8

´penpA´λq ´ vq ´ vV

“ lim supnÑ8

enpA´λqvV “ 0.(4.92)

Lemma 4.5.13 hence proves for all λ P p0,8q, v P V that Rλv P DpAq and

pλ´ AqRλv “ pλ´ Aq

ˆ

limnÑ8

ż n

0

espA´λqv ds

˙

“ limnÑ8

„

pλ´ Aq

ˆż n

0

espA´λqv ds

˙

“ v.

(4.93)

Moreover, Lemma 4.5.7, the fundamental theorem of calculus, and (4.91) show forall λ P p0,8q, v P DpAq that

Rλpλ´ Aqv “ limnÑ8

ż n

0

espA´λqpλ´ Aqv ds “ ´ limnÑ8

ż n

0

espA´λqpA´ λqv ds

“ limnÑ8

pv ´ enpA´λqvq “ v.(4.94)

In addition, note that (4.89) proves for all λ P p0,8q that RλLpV q ď1λ. This,

(4.93), and (4.94) prove that p0,8q Ď ρpAq, that it holds for all λ P p0,8q thatpλ´ Aq´1 “ Rλ, and that

suphPp0,8q

›

›p1´ hAq´1›

›

LpV q“ sup

λPp0,8q

›

›p1´ 1λ ¨ Aq´1›

›

LpV q“ sup

λPp0,8q

›

›λpλ´ Aq´1›

›

LpV qď 1.

(4.95)


The proof of the “ñ” direction in the statement of Theorem 4.7.8 is thereby com-pleted. It thus remains to prove the “ð” direction in the statement of Theorem 4.7.8.To this end let pAλqλPρpAq Ď LpV q be the family of Yosida approximations of A andassume for the rest of this proof that DpAq is dense in V , that A is a closed linearoperator, that p0,8q Ď ρpAq, and that suphPp0,8q p1´ hAq

´1LpV q ď 1. Lemma 4.7.7

implies for all λ P p0,8q that petAλqtPr0,8q Ď LpV q is a uniformly continuous contrac-tion semigroup. This, again Lemma 4.7.7, and Corollary 4.7.6 prove for all t P r0,8q,v P V , pxnqnPN Ď DpAq with lim supnÑ8 v ´ xnV “ 0 that

limNÑ8

supλ,µPrN,8q

supsPr0,ts

esAλv ´ esAµvV (4.96)

“ lim supnÑ8

limNÑ8

supλ,µPrN,8q

supsPr0,ts

esAλv ´ esAλxn ` esAλxn ´ e

sAµxn ` esAµxn ´ e

sAµvV

ď 2 ¨ lim supnÑ8

v ´ xnV ` t ¨ lim supnÑ8

limNÑ8

supλ,µPrN,8q

Aλxn ´ AµxnV “ 0.

This and the assumption that DpAq is dense in V show for all t P r0,8q, v P V thatpetAnvqnPN is a Cauchy sequence in pV, ¨V q. In addition, observe that it holds forall t P r0,8q, v P V that

›

›limnÑ8 etAnv

›

›

Vď vV limnÑ8 e

tAnLpV q ď vV . (4.97)

For the remainder of this proof let S : r0,8q Ñ LpV q be the function which satisfiesfor all t P r0,8q, v P V that

Stv “ limnÑ8

etAnv. (4.98)

Inequality (4.97) implies that suptPr0,8q StLpV q ď 1. In addition, note that S0 “ IdVand that it holds for all t1, t2 P r0,8q, v P V that

St1St2v ´ St1`t2vV “›

›

›

”

limnÑ8

et1An´

limmÑ8

et2Amv¯ı

´

”

limnÑ8

et1Anet2Anvı›

›

›

V

“ limnÑ8

›

›

›et1An

´”

limmÑ8

et2Amvı

´ et2Anv¯›

›

›

V

ď

›

›

›

”

limmÑ8

et2Amvı

´

”

limnÑ8

et2Anvı›

›

›

V“ 0.

(4.99)

Moreover, (4.96) implies for all t P r0,8q, v P V that

lim supnÑ8

supsPr0,ts

esAnv ´ SsvV “ lim supnÑ8

supsPr0,ts

limmÑ8

esAnv ´ esAmvV

ď lim supNÑ8

supn,mPtN,N`1,...u

supsPr0,ts

esAnv ´ esAmvV “ 0.(4.100)


This ensures for all v P V that

lim suptŒ0

Stv ´ vV “ lim supnÑ8

limtŒ0

supsPr0,ts

Ssv ´ esAnv ` esAnv ´ vV

ď lim supnÑ8

supsPr0,1s

Ssv ´ esAnvV ` lim sup

nÑ8lim suptŒ0

etAnv ´ vV “ 0.(4.101)

Therefore, it holds that S is a strongly continuous contraction semigroup. Nextobserve that Corollary 4.7.6 and (4.100) show for all t P r0,8q, v P DpAq that

lim supnÑ8

›

›

›

›

ż t

0

esAnAnv ds´

ż t

0

SsAv ds

›

›

›

›

V

ď t ¨ lim supnÑ8

supsPr0,ts

esAnAnv ´ SsAvV

“ t ¨ lim supnÑ8

supsPr0,ts

esAnAnv ´ esAnAv ` esAnAv ´ SsAvV (4.102)

ď t ¨ lim supnÑ8

Anv ´ AvV ` t ¨ lim supnÑ8

supsPr0,ts

esAnAv ´ SsAvV “ 0.

The fundamental theorem of calculus and Lemma 4.5.7 hence ensure for all v P DpAqthat

lim suptŒ0

›

›

›

›

Stv ´ v

t´ Av

›

›

›

›

V

“ lim suptŒ0

›

›

1t

`

limnÑ8retAnv ´ vs

˘

´ Av›

›

V

“ lim suptŒ0

›

›

›

›

1

t

ˆ

limnÑ8

ż t

0

esAnAnv ds

˙

´ Av

›

›

›

›

V

“ lim suptŒ0

›

›

›

›

1

t

ˆż t

0

SsAv ds

˙

´ Av

›

›

›

›

V

“ 0.

(4.103)

Therefore, we obtain for all v P DpAq that v P DpGSq and GSv “ Av. This combinedwith the assumption that p0,8q Ď ρpAq shows that p1 ´ Aq : DpAq Ď V Ñ V isbijective and

p1´ GSq`

DpAq˘

“ p1´ Aq`

DpAq˘

“ V. (4.104)

Moreover, the “ñ” direction in the statement of Theorem 4.7.8 proves that p1 ´GSq : DpGSq Ď V Ñ V is bijective. This and (4.104) imply that DpGSq “ p1 ´GSq´1pXq “ DpAq. Equality (4.103) hence ensures that GS “ A. The proof ofTheorem 4.7.8 is thus completed.

4.8. SEMIGROUPS GENERATED BY DIAGONAL OPERATORS 125

Remark 4.7.9 (Boundedness of implicit Euler approximations for linear differentialequations). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A : DpAq ĎV Ñ V be a linear operator such that p0,8q Ď ρpAq and suphPp0,8q p1´ hAq

´1LpV q ď

1, and let pY hn qnPN0 Ď V , h P p0,8q, satisfy for all h P p0,8q, n P N0 that

Y hn`1 “ p1´ hAq

´1Y hn . (4.105)

Then it holds for all h P p0,8q, n P N0 that

Y hn`1V ď

„

suprPp0,8q

›

›p1´ rAq´1›

›

LpV q

Y hn V ď Y

hn V . (4.106)

4.8 Semigroups generated by diagonal operators

Lemma 4.8.1. It holds for all z P Czt0u that

ˇ

ˇ

ez´1z

ˇ

ˇ ď?

2` emaxtRepzq,0u. (4.107)

Proof of Lemma 4.8.1. First, observe that for all a, b P R with a2 ` b2 ą 0 it holdsthat

ˇ

ˇeaìb ´ 1ˇ

ˇ

?a2 ` b2

ď

ˇ

ˇeaìb ´ eibˇ

ˇ`ˇ

ˇeib ´ 1ˇ

ˇ

?a2 ` b2

“|ea ´ 1|?a2 ` b2

`

ˇ

ˇeib ´ 1ˇ

ˇ

?a2 ` b2

“emaxta,0u

ˇ

ˇea´maxta,0u ´ e´maxta,0uˇ

ˇ

?a2 ` b2

`

«

|cospbq ´ 1|2 ` |sinpbq|2

a2 ` b2

ff12

.

(4.108)

The fact that @ b P R : | sinpbq| ď |b|, the fact that @ b P R : |1ćospbq| “ 1ćospbq ď|b|, and the fact that @x P p´8, 0s : |ex ´ 1| ď |x| hence show that for all a, b P Rwith a2 ` b2 ą 0 it holds that

ˇ

ˇeaìb ´ 1ˇ

ˇ

?a2 ` b2

ď|a| emaxta,0u

?a2 ` b2

`

„

b2 ` b2

a2 ` b2

12

“|a| emaxta,0u `

?2 |b|

?a2 ` b2

ď?

2` emaxta,0u.

(4.109)



Theorem 4.8.2 (Semigroups generated by diagonal operators*). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let B Ď H be an orthonormal basis, letλ : B Ñ K be a function which satisfies that supptRepλbq : b P Bu Y t0uq ă 8, andlet A : DpAq Ď H Ñ H be the linear operator which satisfies for all v P DpAq that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉|2 ă 8

+

(4.110)

and Av “ř


(i) it holds that A is a generator of a strongly continuous semigroup,

(ii) it holds for all v P H, t P r0,8q that

eAtv “ÿ

bPB

eλbt 〈b, v〉H b, (4.111)

and

(iii) it holds for all t P r0,8q that eAt P LpHq is a diagonal linear operator.

Proof of Theorem 4.8.2*. Throughout this proof assume w.l.o.g. that B ‰ H andlet S : r0,8q Ñ LpHq be the function with the property that for all v P H, t P r0,8qit holds that

Stpvq “ÿ

bPB

eλbt 〈b, v〉H b. (4.112)

Note that the assumption that supbPBRepλbq ă 8 ensures that such a function doesindeed exist. Next observe that for all t1, t2 P r0,8q, v P H it holds that

St1pSt2pvqq “ St1

˜

ÿ

bPB

eλbt2 〈b, v〉H b

¸

“ÿ

bPB

eλbt2 〈b, v〉H St1pbq

“ÿ

bPB

eλbt2 〈b, v〉H

«

ÿ

cPB

eλct1 〈c, b〉H c

ff

“ÿ

bPB

eλbt2 〈b, v〉H“

eλbt1b‰

“ÿ

bPB

eλbpt1`t2q 〈b, v〉H b “ St1`t2pvq.

(4.113)


The function S is thus a semigroup. Moreover, observe that Lebesgue’s theorem ofdominated convergence proves that for all v P H it holds that

lim suptŒ0

Stv ´ v2H “ lim sup

tŒ0

›

›

›

›

›

ÿ

bPB

“

eλbt ´ 1‰

〈b, v〉H b

›

›

›

›

›

2

H

“ lim suptŒ0

«

ÿ

bPB

›

›

“

eλbt ´ 1‰

〈b, v〉H b›

›

2

H

ff

“ lim suptŒ0

«

ÿ

bPB

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2

ff

“ lim suptŒ0

ż

B

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2 #Bpdbq

“

ż

B

lim suptŒ0

´

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2¯

#Bpdbq “ 0.

(4.114)

Combining this with Lemma 4.5.15 proves that S is a strongly continuous semigroup.In the next step we observe that Lemma 4.8.1 ensures that

sup

˜#

ˇ

ˇ

ˇ

ˇ

reλbt ´ 1´ λbts

λbt

ˇ

ˇ

ˇ

ˇ

2

: b P λ´1pCzt0uq, t P p0, 1s

+

Y t0u

¸

ď 2 supzPCzt0u,

RepzqăsupbPBRepλbq

ˇ

ˇ

ˇ

ˇ

ez ´ 1

z

ˇ

ˇ

ˇ

ˇ

2

` 2

ď 2”?

2` exppsupptRepλbq : b P Bu Y t0uqqı2

` 2 ă 8.

(4.115)

This and Lebesgue’s theorem of dominated convergence shows that for all v P DpAq


it holds that

lim suptŒ0

˜

Stv ´ v ´ tAv2H

t2

¸

“ lim suptŒ0

˜

1

t2

ÿ

bPB

›

›

“

eλbt ´ 1´ λbt‰

〈b, v〉H b›

›

2

H

¸

“ lim suptŒ0

¨

˚

˝

ÿ

bPB,λb‰0

ˇ

ˇ

ˇ

ˇ

ˇ

“


λbt

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2

˛

‹

‚

“

ż

λ´1pCzt0uq

lim suptŒ0

ˇ

ˇ

ˇ

ˇ

ˇ

“


λbt

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2 #Bpdbq

“ÿ

bPλ´1pCzt0uq

lim suptŒ0

ˇ

ˇ

ˇ

ˇ

ˇ

8ÿ

k“2

rλbtsk´1

k!

ˇ

ˇ

ˇ

ˇ

ˇ

2

|λb 〈b, v〉H |2

ďÿ

bPB

|λb 〈b, v〉H |2 lim sup

tŒ0

«

|λbt|8ÿ

k“0

|λbt|k

k!

ff2

“ÿ

bPB

|λb 〈b, v〉H |2

„

lim suptŒ0

|λbt|2 e2|λbt|

“ 0.

(4.116)

Hence, we obtain that DpAq Ď DpGSq and GS|DpAq “ A. Next note that for allv P DpGSq it holds that

ÿ

bPB

|λb 〈b, v〉H |2“

ÿ

bPB,λb‰0

|λb 〈b, v〉H |2“

ÿ

bPB,λb‰0

lim inftŒ0

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1

λbt¨ λb ¨ 〈b, v〉H

ˇ

ˇ

ˇ

ˇ

2

“

ż

λ´1pCzt0uq

lim inftŒ0

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1


ˇ

ˇ

ˇ

ˇ

2

#Bpdbq

ď lim inftŒ0

ż

λ´1pCzt0uq

ˇ

ˇ

ˇ

ˇ

eλbt ´ 1


ˇ

ˇ

ˇ

ˇ

2

#Bpdbq

“ lim inftŒ0

˜

1

t2

ÿ

bPB

ˇ

ˇ

“

eλbt ´ 1‰

〈b, v〉Hˇ

ˇ

2

¸

“ lim inftŒ0

˜

Stv ´ v2H

t2

¸

ď lim suptŒ0

˜

Stv ´ v ´ tGSv2Ht2

` GSv2H

¸

“ GSv2H ă 8.

(4.117)

This establishes that DpGSq Ď DpAq. This together with the facts that DpAq Ď


DpGSq and GS|DpAq “ A assures that GS “ A. The proof of Theorem 4.8.2 is thuscompleted.

Proposition 4.8.3 (Semigroups generated by diagonal operators*). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then it holds that supλPσP pAqRepλq ă 8 if and only if A is agenerator of a strongly continuous semigroup.

Proof of Proposition 4.8.3*. First, observe that Theorem 4.8.2 shows that the con-dition supλPσP pAqRepλq ă 8 implies that A is a generator of a strongly continuoussemigroup. In remainder of this proof we thus assume that A is the generator of astrongly continuous semigroup S : r0,8q Ñ LpHq. Note that the assumption that Ais a diagonal linear operator ensures that there exists an orthonormal basis B Ď Hof H and a function λ : BÑ K such that for all v P DpAq it holds that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.118)

andAv “

ÿ

bPB

λb 〈b, v〉H b. (4.119)

Next observe that the fact that GS “ A and Lemma 4.5.7 imply that for all b P B,t P r0,8q, v P H it holds that the function

r0,8q Q s ÞÑ Ssb P H (4.120)

is continuously differentiable, that 〈v, S0pbq〉H “ 〈v, b〉H , and that

ddt〈v, Stpbq〉H “ 〈v,GSStpbq〉H “ λb 〈v, Stpbq〉H . (4.121)

This shows that for all b P B, v P H, t P r0,8q it holds that

〈v, Stb〉H “ eλbt 〈v, b〉H . (4.122)

Hence, we obtain that for all b P B, t P r0,8q it holds that

Stb “ eλbtb. (4.123)

This, in turn, ensures that

8 ą S1LpHq ě supbPB

S1bH “ supbPB

ˇ

ˇeλbˇ

ˇ “ supbPB

ˇ

ˇeRepλbqˇ

ˇ “ esupbPBRepλbq. (4.124)

This implies that supbPBRepλbq ă 8. The proof of Proposition 4.8.3 is thus com-pleted.


4.8.1 Smoothing effect of the semigroup

Proposition 4.8.4 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space,let A : DpAq Ď H Ñ H be a symmetric linear operator with infpσP pAqq ą 0, letB Ď H be an orthonormal basis, let λ : B Ñ K be a function such that for allv P DpAq it holds that

DpAq “

#

w P H :ÿ

bPB

|λb 〈b, w〉H |2ă 8

+

(4.125)

and Av “ř

bPB λb 〈b, v〉H b, and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of inter-polation spaces associated to A. Then

• it holds that B Ď pXrPRHrq,

• it holds for all r P R that spanpBqHr“ Hr, and

• it holds for all r P R that A´rpBq “

bpλbqr

P H : b P B(

is an orthonormalbasis of Hr.

Proof of Proposition 4.8.4*. Observe that Proposition 4.8.4 follows immediately fromDefinition 3.6.21, Definition 3.6.22, and Definition 3.6.30.

In the next result, Theorem 4.8.5, we establish a smoothing effect for strongly con-tinuous semigroups generated by diagonal linear operators. We recall Remark 2.4.24for the formulation of Theorem 4.8.5.


Theorem 4.8.5 (Smoothing effect of semigroups generated by diagonal operators*).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ Hbe a symmetric linear operator with suppσP pAqq ă 0, let B Ď H be an orthonormalbasis, let λ : BÑ K be a function such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.126)


Av “ÿ

bPB

λb 〈b, v〉H b, (4.127)

and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated toÁ. Then

(i) it holds that for all r P r0,8q that

suptPr0,8q

›

›p´tAqreAt›

›

LpHqď

”r

e

ır

ă 8, (4.128)

(ii) it holds for all t P p0,8q, r P p´8, 0q, v P H that eAtvH ď |r||r| tr vHr ă 8,

(iii) it holds for all t P p0,8q, r P R that

eAtpHrq Ď pXsPRHsq (4.129)

(cf. Proposition 2.4.22 and Item (ii)),

(iv) it holds for all t P r0,8q, r P p´8, 0q, v P H that etAvHr ď vHr , and

(v) it holds for all t P r0,8q, v P pYrPRHrq that

eAtv “ÿ

bPB

eλbt 〈b, v〉H b (4.130)

(cf. Proposition 2.4.22 and Item (iv)).

Proof of Theorem 4.8.5*. Observe that Proposition 4.8.4 implies that for all r Pr0,8q it holds that

suptPr0,8q

›

›p´tAqreAt›

›

LpHq“ sup

tPr0,8q

supbPB

ˇ

ˇp´tλbqreλbt

ˇ

ˇ ď supxPp0,8q

„

xr

ex

ď

”r

e

ır

ă 8.

(4.131)The proof of Theorem 4.8.5 is thus completed.


Lemma 4.8.6 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 0.Then

suprPr0,1s

suptPp0,8q

›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHqď 1. (4.132)

Proof of Lemma 4.8.6*. Observe that for all t P p0,8q, r P r0, 1s it holds that

›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHq“ sup

λPσP ptAqq

ˇ

ˇp´λq´r`

eλ ´ 1˘ˇ

ˇ

ď supxPp0,8q

„

p1´ e´xq

xr

ď 1.(4.133)


Exercise 4.8.7 (*). Let T P p0,8q, r P r0, 1q, K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be aK-Hilbert space, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operator withsup

`

σP pAq˘

ă 0, and let e : r0, T s Ñ H be a continuous function with the property

that for all t P r0, T s it holds that eptq “ ep0q `şt

0pÁqr ept´sqA epsq ds. Prove that

suptPr0,T s eptqH ď ep0qH ¨ E1´r

“

T 1´r‰

.

4.8.2 Semigroup generated by the Laplace operator

Example 4.8.8 (Heat equation with Dirichlet boundary conditions*). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq and let v : p0, 1q Ñ R be a twice continuouslydifferentiable function with vp0`q “ vp1´q “ 0. Then

(i) it holds that supλPσP pAq λ “ ´π2 ă 8,


(ii) it holds that A is the generator of a strongly continuous semigroup (cf. Theo-rem 4.8.2 and Item (i)), and

(iii) it holds that the function u : r0,8q ˆ p0, 1q Ñ R with @ pt, xq P r0,8q ˆp0, 1q : upt, xq “ peAtvqpxq satisfies for all pt, xq P p0,8qˆp0, 1q that u|p0,8qˆp0,1q PC2pp0,8q ˆ p0, 1q,Rq and

B

Btupt, xq “ B2

Bx2upt, xq, up0, xq “ vpxq, upt, 0`q “ upt, 1´q “ 0 (4.134)

(cf. Theorem 4.8.5, Item (ii), and Lemma 4.5.7).

Class exercise 4.8.9 (Laplacian on L2pBorelp0,1q; |¨|Rq). Let A : DpAq Ď L2pBorelp0,1q;|¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator which satisfies for all v P H2pp0, 1q,Rq

that DpAq “ H2pp0, 1q,Rq and Av “ v2. Is A a generator of a strongly continuoussemigroup?

Example 4.8.10 (Laplacian on L2pBorelp0,1q; |¨|Rq). Let A : DpAq Ď L2pBorelp0,1q; |¨|RqÑ L2pBorelp0,1q; |¨|Rq be the linear operator which satisfies for all v P H2pp0, 1q,Rqthat DpAq “ H2pp0, 1q,Rq and Av “ v2. Then

(i) it holds that rp1qxPp0,1qsBorelp0,1q,BpRq P Kernp0´ Aq “ KernpAq,

(ii) it holds that 0 P σP pAq,

(iii) it holds for all n P N, x P p0, 1q that d2

dx2sinpnπxq “ ń2π2 sinpnπxq,

(iv) it holds for all n P N that rpsinpnπxqqxPp0,1qsBorelp0,1q,BpRq P Kernpń2π2 ´ Aq,

(v) it holds for all n P N that ń2π2 P σP pAq,

(vi) it holds thatş1

01 ¨ sinpπxq dx “ r´ cospπxqsx“1

x“0 “ 1´ cospπq ‰ 0,

(vii) it does not hold that for all v P Kernp0 ´ Aq, w P Kernp´π2 ´ Aq it holds thatş1

0vpxqwpxq dx “ 0, and

(viii) it holds that A is not a diagonal linear operator (cf. Proposition 3.6.16 andDefinition 3.6.1).


Part II

Foundations in probability theory

135

Chapter 5

Random variables with values ininfinite dimensional spaces

In the most of this chapter we follow the presentations in Da Prato & Zabczyk [3],Werner [23] and Prevot & Rockner [18].

5.1 General measure and probability spaces

5.1.1 Uniqueness theorem for measures

Definition 5.1.1 (Image measure/Pushforward measure*). Let pΩ,A, µq be a mea-sure space, let pΩ, Aq be a measurable space, and let f : Ω Ñ Ω be an A/A-measurablemapping. Then we denote by fpµqA : AÑ r0,8s the function with the property thatfor all A P A it holds that

`

fpµqA˘

pAq “ µ`

f´1pAq

˘

(5.1)

and we call fpµqA the image measure associated to f and A.

Definition 5.1.2 (X-Stability). Let Ω be a set and let E Ď PpΩq be a set with theproperty that for all a, b P E it holds that aX b P E. Then E is called X-stable.

Definition 5.1.3 (λ-system). Let Ω be a set and let A P PpPpΩqq be such that

(i) it holds that Ω P A,

(ii) it holds for all A,B P A with A Ď B that BzA P A, and

(iii) it holds for all pairwise disjoint sets pAnqnPN Ď A that YnPNAn P A.

Then and only then we say that A is a λ-system.

137

138 CHAPTER 5. RANDOM VARIABLES

Definition 5.1.4. Let Ω be a set and let A P PpPpΩqq. Then we denote by δΩpAqthe set given by

δΩpAq “ X B is a λ-systemwith AĎBĎPpΩq

B (5.2)

and we call δΩpAq the λ-system generated by A.

Lemma 5.1.5. Let Ω be a set and let C P PpPpΩqq be a λ-system. Then it holdsthat C is X-stable if and only if C is a sigma-algebra on Ω.

Proof of Lemma 5.1.5. Throughout this proof let A P PpPpΩqq be a X-stable λ-system. First of all, note that Ω P A. Next, the assumption that A is a λ-systemensures for all A P A that pΩzAq P A. Moreover, the fact that @A,B P A : AXB P Aimplies that for all A,B P A it holds that AzB “ pAzpAXBqq P A. In the next stepnote that for all pAnqnPN Ď A it holds that

YnPNAn “ A1 Y`

YnPN pAn`1zpYmPt1,2,...,nuAmqq˘

P A. (5.3)

The fact that every sigma-algebra on Ω is X-stable thus completes the proof ofLemma 5.1.5.

Theorem 5.1.6 can, e.g., be found as Lemma 1.42 in Klenke [13].

Theorem 5.1.6. Let Ω be a set and let A P PpPpΩqq be X-stable. Then σΩpAq “δΩpAq.

Proof of Theorem 5.1.6. Throughout this proof we denote by DB Ď PpΩq, B P

δΩpAq, the sets with the property that for all B P δΩpAq it holds that DB “ tA PδΩpAq : AXB P δΩpAqu. Note that for all B P δΩpAq it holds that ΩXB “ B P δΩpAq.This proves that for all B P δΩpAq it holds that Ω P DB. In the next step observe thatfor all A P δΩpAq, B,C P DA with B Ď C it holds that pCzBqXA “ pCXAqzpBXAq PδΩpAq. Moreover, for all A P δΩpAq and all pairwise disjoint sets pAnqnPN Ď DA itholds that

pYnPNAnq X A “ YnPNpAn X Aq P δΩpAq. (5.4)

This proves that for all A P δΩpAq it holds that DA is a λ-system. In the next stepnote that the assumption that A is X-stable implies for all A P A that A Ď DA.Hence, we conclude that for all A P A it holds that δΩpAq Ď δΩpDAq “ DA. Thisimplies that for all A P A, B P δΩpAq it holds that AXB P δΩpAq. Furthermore, thisensures that for all A P A, B P δΩpAq it holds that A P DB. Moreover, this showsthat for all B P δΩpAq it holds that A Ď DB. Hence, for all B P δΩpAq it holds that

5.1. GENERAL MEASURE AND PROBABILITY SPACES 139

δΩpAq Ď δΩpDBq “ DB. Finally, this proves that for all A,B P δΩpAq it holds thatA P DB. This hence shows that δΩpAq is X-stable. The proof of Theorem 5.1.6 isthus completed.

Theorem 5.1.7 can, e.g., be found as Lemma 1.42 in Klenke [13].

Theorem 5.1.7 (Uniqueness theorem for measures). Let Ω be a set, let E Ď PpΩqbe a X-stable subset of PpΩq, let µ1, µ2 : σΩpEq Ñ r0,8s be measures which satisfythat there exists a sequence Ωn P tA P E : µ1pAq ă 8u, n P N, such that YnPNΩn “ Ωand µ1|E “ µ2|E . Then µ1 “ µ2.

Proof of Theorem 5.1.7. Throughout this proof let S Ď E and DE Ď σΩpEq, E P E ,be the sets which satisfy for all E P S that S “ tA P E : µ1pAq ă 8u and DE “ tA PσΩpEq : µ1pAXEq “ µ2pAXEqu. First, note that for all E P S it holds that Ω P DE.Next observe that for all E P S, A,B P DE with B Ď A it holds that

µ1ppAzBq X Eq “ µ1pAX Eq ´ µ1pB X Eq

“ µ2pAX Eq ´ µ2pB X Eq “ µ2ppAzBq X Eq.(5.5)

This shows for all E P S, A,B P DE with B Ď A that AzB P DE. Moreover, notethat for all E P S, pAnqnPN Ď DE with @ i P N, j P Nztiu : AiXAj “ H it holds that

µ1

´

`

ď

nPN

An˘

X E¯

“

8ÿ

n“1

µ1pAn X Eq “8ÿ

n“1

µ2pAn X Eq “ µ2

´

`

ď

nPN

An˘

X E¯

.

(5.6)

This proves for all E P S, pAnqnPN Ď DE with @ i P N, j P Nztiu : Ai X Aj “ H thatYnPNAn P DE. Hence, we have established that for all E P S it holds that DE isa λ-system. The fact that for all E P S it holds that E Ď DE hence shows for allE P S that δΩpEq Ď DE. Combining this, the assumption that E is X-stable, andTheorem 5.1.6 ensures for all E P S that σΩpEq “ δΩpEq Ď DE Ď σΩpEq. This assuresfor all E P S that

σΩpEq “ DE. (5.7)

Finally, this hence proves for all A P σΩpEq that

µ1pAq “ limnÑ8 µ1pAX Ωnq “ limnÑ8 µ2pAX Ωnq “ µ2pAq. (5.8)



Corollary 5.1.8 (Uniqueness theorem for finite measures). Let Ω be a set, let E ĎPpΩq be a X-stable subset of PpΩq, let µ1, µ2 : σΩpEq Ñ r0,8s be finite measureswhich satisfy that µ1|EYtΩu “ µ2|EYtΩu. Then µ1 “ µ2.

Proof of Corollary 5.1.8. Throughout this proof let F be the set given by F “

E Y tΩu. Next observe that the fact that Ω P F “ tA P F : µ1pAq ă 8u andTheorem 5.1.7 (with E “ F and Ωn “ Ω P tA P F : µ1pAq ă 8u for n P N in thenotation of Theorem 5.1.7) ensure that µ1 “ µ2. The proof of Corollary 5.1.8 is thuscompleted.

5.1.2 Independence on probability spaces

The next result, Lemma 5.1.9, provides a characterization for independence. Theproof of Lemma 5.1.9 is similiar to the proof of Proposition 2.2.5 in Chapter 2.

Lemma 5.1.9 (A characterization for independence). Let pΩ,F ,Pq be a probabilityspace, let I be a set, let pEi, diq, i P I, be a family of metric spaces, and for everyi P I let Xi : Ω Ñ Ei be an F/BpEiq-measurable mapping. Then the following fourstatements are equivalent:

(i) The family Xi, i P I, is independent.

(ii) It holds for every family ϕi PMpBpEiq,BpRqq, i P I, that the family ϕi ˝ Xi,i P I, is independent.

(iii) It holds for every family ϕi P CpEi,Rq, i P I, that the family ϕi ˝Xi, i P I, isindependent.

(iv) It holds for every family ϕi P CpEi,Rq, i P I, with @ i P I : suppt|ϕipxq| : x PEiu Y t0uq ă 8 that the family ϕi ˝Xi, i P I, is independent.

Proof of Lemma 5.1.9. First, note that it is clear that p(i) ñ (ii)q, that p(ii) ñ (iii)q,and that p(iii) ñ (iv)q. It thus remains to prove that p(iv) ñ (i)q. For this let J Ď Ibe a finite set, let Ai P BpEiq, i P J , be a family which satisfies @ i P J : Ai ‰ H, andlet ϕi P CpEi, Rq, i P J , be the family which satisfies for all i P J , x P Ei that

ϕipxq “ min

1, distdiptxu, EizAiq(

. (5.9)

Next observe that Item (iv) ensures that

P`

XiPJ

Xi P Ai(˘

“ P`

XiPJ

Xi P pϕiq´1pp0,8qq

(˘

“ P`

XiPJ

ϕipXiq P p0,8q(˘

“ś

iPJ P`

ϕipXiq P p0,8q˘

“ś

iPJ P`

Xi P pϕiq´1pp0,8qq

˘

“ś

iPJ P`

Xi P Ai˘

.

(5.10)

5.1. GENERAL MEASURE AND PROBABILITY SPACES 141

This establishes that p(iv) ñ (i)q. The proof of Lemma 5.1.9 is thus completed.

Rouhgly speaking, the next result, Lemma 5.1.10, demonstrates that it inde-pendence of a family of X-stable generating systems ensures that the family of thecorresponding generated sigma-algebras is independent. Lemma 5.1.10 can, e.g., befound as Item (iii) in Theorem 2.13 in Klenke [13].

Lemma 5.1.10 (Independence of X-stable sets*). Let pΩ,F ,Pq be a probabilityspace, let I be a set, let Ai P PpFq, i P I, be a P-independent family, and as-sume for every i P I that pAi Y tHuq is X-stable. Then it holds that the familyσΩpAiq P PpFq, i P I, is P-independent.

5.1.3 Factorization lemma for conditional expectations

Lemma 5.1.11. Let pΩ,Fq be a measurable space and let f P MpF ,Bpr0,8sqq.Then there exists a sequence fn : Ω Ñ r0,8q, n P N, of simple functions with theproperty that for all n P N, ω P Ω it holds that fnpωq ď fn`1pωq and limmÑ8 fmpωq “fpωq.

Proof of Lemma 5.1.11. Throughout this proof let An, An,j P PpΩq, n P N, j PN X r1, n2ns, be the sets with the property that for all n P N, j P N X r1, n2ns itholds that An “ tω P Ω: fpωq P rn,8su and An,j “ tω P Ω: fpωq P rpj´1q2n, j2nqu,and let fn : Ω Ñ r0,8q, n P N, be the functions with the property that for all n P N,ω P Ω it holds that

fnpωq “ n1Anpωq `n2nÿ

j“1

j´12n1An,jpωq. (5.11)

First of all note that for every ω P Ω with fpωq P r0,8q it holds that there existj, n P N such that fpωq P rpj´1q2n, j2nq. This and (5.11) imply that for every ω P Ωwith fpωq P r0,8q it holds that there exists n P N such that 0 ď fpωq´fnpωq ă 2ń.This hence shows that for all ω P Ω with fpωq P r0,8q it holds that limnÑ8 fnpωq “fpωq. In addition, note that for all ω P Ω with fpωq “ 8 it holds that for all m P N

it holds that fmpωq “ m and ω P XnPNAn. This hence proves that for all ω P Ω withfpωq “ 8 it holds that limnÑ8 fnpωq “ 8. The fact that for all n P N, ω P Ω itholds that fnpωq ď fn`1pωq completes the proof of Lemma 5.1.11.

Lemma 5.1.12 is well known (cf., e.g., Proposition 1.12 in Da Prato & Zabczyk [3]).For completeness the proof of Lemma 5.1.12 is given below.


Lemma 5.1.12. Let pΩ,F ,Pq be a probability space, let pD,Dq and pE, Eq be mea-surable spaces, let X ,Y P PpFq be independent sigma-algebras, let X P MpX ,Dq,Y PMpY , Eq, Φ PMpDbE ,Bpr0,8sqq, Ψ PMpD, r0,8sq, and assume for all x P Dthat Φpx, Y q P L1pP;Rq, ΦpX, Y q P L1pP;Rq, and Ψpxq “ ErΦpx, Y qs. Then

ErΦpX, Y q|Xs “ ErΦpX, Y q|X s “ rΨpXqsP,Bpr0,8sq. (5.12)

Proof of Lemma 5.1.12. Throughout this proof let φn : D ˆ E Ñ r0,8q, n P N, bea sequence of simple functions with the property that for all m P N, x P D, y P E itholds that φmpx, yq ď φm`1px, yq and limnÑ8 φnpx, yq “ Φpx, yq, let ψn : D Ñ r0,8s,n P N, be the functions with the property that for all n P N, x P D it holds thatψnpxq “ Erφnpx, Y qs, and let γA : D Ñ r0,8s, A P D b E , be the functions with theproperty that for all A P D b E , x P D it holds that γApxq “ Er1Apx, Y qs. First ofall note that for all A P D, B P E it holds that

Er1AˆBpX, Y q|X s “ Er1ApXq1BpY q|X s “ r1ApXqsP,Bpr0,8qqEr1BpY q|X s. (5.13)

Combining this with Theorem 5.1.6 assures that for all A P D b E it holds that

Er1ApX, Y q|X s “ rγApXqsP,Bpr0,8qq. (5.14)

Next note that the monotone convergence theorem ensures for all x P D that

limnÑ8 ψnpxq “ limnÑ8Erφnpx, Y qs “ ErlimnÑ8 φnpx, Y qs “ ErΦpx, Y qs “ Ψpxq.(5.15)

Combining the conditional theorem of monotone convergence, the linearity of theconditional expectation, (5.14), and this implies that

ErΦpX, Y q|X s “ ErlimnÑ8 φnpX, Y q|X s “ limnÑ8ErφnpX, Y q|X s“ limnÑ8rψnpXqsP,Bpr0,8sq “ rlimnÑ8 ψnpXqsP,Bpr0,8sq “ rΨpXqsP,Bpr0,8sq.

(5.16)

Moreover, the tower property hence proves that

ErΦpX, Y q|Xs “ E“

ErΦpX, Y q|X s|X‰

“ E“

ΨpXqˇ

ˇX‰

“ rΨpXqsP,Bpr0,8sq. (5.17)


5.2. BOREL SIGMA-ALGEBRAS ON NORMED VECTOR SPACES 143

5.2 Borel sigma-algebras on normed vector spaces

5.2.1 The Hahn-Banach theorem

We first recall the Hahn-Banach theorem (see, e.g., Theorem III.1.5 in Werner [23]).

Theorem 5.2.1 (Hahn-Banach theorem; Extension of continuous linear function-als*). Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, let U Ď V be aK-subspace of V , and let φ P U 1. Then there exists a ϕ P V 1 with the property that

ϕ|U “ φ and ϕV 1 “ φU 1 . (5.18)

The proof of Theorem 5.2.1 uses the axiom of choice. The next corollary is animmediate consequence of the Hahn-Banach theorem.

Corollary 5.2.2 (Projections into 1-dimensional subspaces*). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space with V ‰ t0u, and let v P V . Then there existsa ϕ P V 1 with the property that

ϕpvq “ vV and ϕV 1 “ 1. (5.19)

Proof of Corollary 5.2.2*. We show Corollary 5.2.2 in two steps. In the first step weassume that v ‰ 0. Let U be the K-subspace of V given by U “ tλv P V : λ P Ku “spantvu and let φ : U Ñ K be the mapping with the property that for all λ P K itholds that

φpλvq “ λ vV . (5.20)

Theorem 5.2.1 implies the existence of a ϕ P V 1 with the property that

ϕ|U “ φ and ϕV 1 “ φU 1 “ 1. (5.21)

This proves (5.19) in the case v ‰ 0. In the second step we assume that v “ 0. Theassumption that V ‰ t0u then shows that there exists u P V with the property thatu ‰ 0. The first step hence proves that there exists a ϕ P V 1 with the property that

ϕpuq “ uV and ϕV 1 “ 1. (5.22)

In addition, observe that ϕpvq “ ϕp0q “ 0 “ vV . The proof of Corollary 5.2.2 isthus completed.


5.2.2 Norm representations in normed vector spaces

The next result, Corollary 5.2.3, is an immediate consequence of Corollary 5.2.2above.

Corollary 5.2.3 (Norm via the dual space*). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space with V ‰ t0u, and let v P V . Then

vV “ supϕPV 1zt0u

ϕpvq

ϕV 1“ sup

ϕPV 1zt0u

|ϕpvq|

ϕV 1. (5.23)

If the normed vector space in Corollary 5.2.3 is separable, then the followingresult, Corollary 5.2.4, can be obtained. Corollary 5.2.4 is also an immediate conse-quence of Corollary 5.2.2 above.

Corollary 5.2.4 (Norm of a separable normed vector space via the dual space*).Let K P tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then thereexists a sequence pϕnqnPN Ď V 1 with the property that for all v P V it holds that

vV “ supnPN

ϕnpvq “ supnPN

|ϕnpvq| . (5.24)

Proof of Corollary 5.2.4*. W.l.o.g. we assume that V ‰ t0u. The assumption thatpV, ¨V q is separable implies that there exists a sequence vn P V , n P N, with theproperty that the set tvn : n P Nu is dense in V . Corollary 5.2.2 hence shows thatthere exists a sequence ϕn P V

1, n P N, with the property that for all n P N it holdsthat

ϕnpvnq “ vnV and ϕnV 1 “ 1. (5.25)

This implies that for all k P N it holds that

vkV “ supnPN

ϕnpvkq. (5.26)

Next let v P V and ε P p0,8q be arbitrary. Then observe that

supnPN

ϕnpvq ď supnPN

rϕnV 1 vV s “ vV . (5.27)


vV ď ε` supnPN

ϕnpvq. (5.28)

5.2. BOREL SIGMA-ALGEBRAS ON NORMED VECTOR SPACES 145

To see this observe that the fact that tvk P V : k P Nu is dense in V ensures thatthere exists a k P N such that v ´ vkV ď

ε2. This implies that

vV ď vkV ` v ´ vkV “ ϕkpvkq ` v ´ vkV“ ϕkpvq ` v ´ vkV ` ϕkpvk ´ vq

ď ϕkpvq ` v ´ vkV ` ϕkV 1 v ´ vkV“ ϕkpvq ` 2 v ´ vkV ď sup

nPNϕnpvq ` 2 v ´ vkV ď sup

nPNϕnpvq ` ε.

(5.29)


5.2.3 Linear characterization of the Borel sigma-algebra

Proposition 5.2.5 (Linear characterization of the Borel sigma-algebra*). Let K P

tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then there exists asequence ϕn P V

1, n P N, such that

BpV q “ σV ppϕqϕPV 1q “ σV pϕ : ϕ P V 1q “ σV pϕn : n P Nq . (5.30)

Proof of Proposition 5.2.5*. Let fv : V Ñ r0,8q, v P V , be the functions with theproperty that for all x, v P V it holds that

fvpxq “ x´ vV . (5.31)

Observe thatBpV q “ σV pfv : v P V q . (5.32)

Next observe that Corollary 5.2.4 implies that there exists a sequence ϕn P V1, n P N,

such that for all v P V it holds that

vV “ supnPN

ϕnpvq. (5.33)

This implies that

σV pϕn : n P Nq “ σV`

ϕnpp¨q ` vq : n P N, v P V˘

Ě σV pfv : v P V q . (5.34)



5.3 Probability measures on normed vector spaces

5.3.1 Fourier transform of a measure

5.3.1.1 Characteristic functionals

Theorem 5.1.7 is, for example, established as Lemma 1.42 in Klenke [14].

Proposition 5.3.1 (Characteristic function). Let d P N and let µk : BpRdq Ñ r0,8s,k P t1, 2u, be finite measures with the property that for all ξ P Rd it holds that

ż

Rd

ei〈ξ,x〉Rd µ1pdxq “

ż

Rd

ei〈ξ,x〉Rd µ2pdxq. (5.35)

Then µ1 “ µ2.

Proposition 5.3.1 is, for example, proved as Theorem 15.8 in Klenke [13].

Definition 5.3.2 (Characteristic functional*). Let pV, ¨V q be a normed R-vectorspace. Then we denote by

FV :

µ PMpBpV q, r0,8sq : µ is a finite measure on pV,BpV qq(

ÑMpV 1,Cq(5.36)

the mapping with the property that for all µ P DpFV q, ϕ P V1 it holds that

pFV µqpϕq “`

FV pµq˘

pϕq “

ż

V

ei¨ϕpxq µpdxq (5.37)

and for every µ P DpFV q we call FV pµq the characteristic functional of µ.

Lemma 5.3.3 (Elementary properties of the characteristic functionals). Let pV, ¨V qbe a normed R-vector space. Then

(i) it holds for all µ P DpFV q that pFV µqp0q “ µpV q,

(ii) it holds for all µ1, µ2 P DpFV q, a P r0,8q that FV paµ1 ` µ2q “ aFV pµ1q `

FV pµ2q, and

(iii) it holds that impFV q Ď CpV 1,Cq.

5.3. PROBABILITY MEASURES ON NORMED VECTOR SPACES 147

Proof of Lemma 5.3.3. First of all, observe that for all µ P DpFV q it holds that

pFV µqp0q “

ż

V

ei0 µpdxq “

ż

V

1µpdxq “ µpV q. (5.38)

Next note that for all µ1, µ2 P DpFV q, a P r0,8q, ϕ P V1 it holds that

`

FV paµ1 ` µ2q˘

pϕq “

ż

V

eiϕpxq ra ¨ µ1pdxq ` µ2pdxqs

“ a

ż

V

eiϕpxqµ1pdxq `

ż

V

eiϕpxqµ2pdxq

“ aFV pµ1q ` FV pµ2q.

(5.39)

Finally, observe that Lebesgue’s theorem of dominated convergence proves that forall µ P DpFV q and all ϕn P V

1, n P N0, with limnÑ8 ϕn ´ ϕ0V 1 “ 0 it holds that

limnÑ8

|pFV µqpϕnq ´ pFV µqpϕ0q| ď limnÑ8

ż

V

ˇ

ˇeiϕnpxq ´ eiϕ0pxqˇ

ˇµpdxq

“

ż

V

limnÑ8

ˇ

ˇeiϕnpxq ´ eiϕ0pxqˇ

ˇµpdxq “ 0.

(5.40)


5.3.1.2 Fourier transform on separable normed vector spaces

Lemma 5.3.4 (Characteristic functional determines measure uniquely). Let pV, ¨V qbe a separable normed R-vector space. Then FV is injective.

Proof of Lemma 5.3.4. Let µ1, µ2 P DpFV q satisfy FV pµ1q “ FV pµ2q. Then notethat for all n P N, φ “ pφ1, . . . , φnq P LpV,R

nq, ξ P Rn it holds thatż

Rn

ei〈ξ,x〉Rn`

φpµ1q˘

pdxq “

ż

V

ei〈ξ,φpvq〉Rn pµ1qpdvq “`

FV µ1

˘

´

xξ, φp¨qyRn¯

“`

FV µ2

˘

´

xξ, φp¨qyRn¯

“

ż

V

ei〈ξ,φpvq〉Rn pµ2qpdvq “

ż

Rn

ei〈ξ,x〉Rn`

φpµ2q˘

pdxq.

(5.41)

Proposition 5.3.1 hence implies that for all n P N, φ P LpV,Rnq it holds that

φpµ1q “ φpµ2q. (5.42)

In the next step let E Ď PpV q be the set given by

E “ď

nPN

φ´1pBq P PpV q : φ P LpV,Rn

q, B P BpRnq(

. (5.43)


Note that E Ď BpV q and observe that (5.42) shows that

µ1|E “ µ2|E . (5.44)

This, the fact that E is X-stable, the fact V P E , and Theorem 5.1.7 imply that

µ1|σV pEq “ µ2|σV pEq. (5.45)

Moreover, observe that Proposition 5.2.5 proves that

σV pEq “ BpV q. (5.46)

Combining this with (5.45) completes the proof of Lemma 5.3.4.

5.3.1.3 Almost surely separably supported

In Theorem 5.3.25 below we prove a generalization of Lemma 5.3.4. For this weneed a few preparations. These preparations and Theorem 5.3.25 are based on thepresentations in Van Neerven [21].

Lemma 5.3.5. Let pE, Eq be a topological space and let A Ď E be separable. ThenA is separable too.

Lemma 5.3.6. Let pE, Eq be a topological space and let A Ď E and B Ď E beseparable. Then AYB is separable too.

Lemma 5.3.7. Let pV, ¨V q be a normed vector space and let A Ď V be separable.Then spanpAq is separable too.

Definition 5.3.8 (Support of a measure). Let pE, Eq be a topological space and letµ : BpEq Ñ r0,8s be a measure on pE,BpEqq. Then we denote by supppµq the setgiven by

supppµq “ tx P E : p@U P E : x P U ñ µpUq ą 0qu (5.47)

and we call supppµq the support of µ.

Class exercise 5.3.9. Let x P R. What is supp`

δRx |BpRq˘

?

Class exercise 5.3.10. Let d P N. What is supppλRdq?

Exercise 5.3.11. Let pE, Eq be a topological space and let µ : BpEq Ñ r0,8s be ameasure on pE,BpEqq. Prove then that supppµq is a closed set in pE, Eq, i.e., provethat Ez supppµq P E.


Remark 5.3.12. In general it is not true that µ`

Ez supppµq˘

“ 0. (see Wikipedia:support (measure theory)).

Definition 5.3.13 (Almost surely separably supported). Let pE, Eq be a topologicalspace and let µ : BpEq Ñ r0,8s be a measure with the property that there exists aseparable and closed subset A Ď E of E such that µpEzAq “ 0. Then µ is called a.s.separably supported (almost surely separably supported).

5.3.1.4 Trace set

Let pΩ,Fq be a measurable space (i.e., Ω is a set and F is a sigma-algebra on Ω)and let A Ď Ω be a subset of Ω. In some situations we are interested to have anappropriate measurable structure (an appropriate sigma-algebra) on A too. Thetrace set in the following definition provides an appropriate concept for this issue;see Lemma 5.3.15 below.

Definition 5.3.14 (Trace set). Let A and Ω be sets and let A Ď PpΩq be a subsetof the power set of Ω. Then we denote by A \A the set given by

A \A “ tAXB P PpAq : B P Au (5.48)

and we call A \A the trace set (of A in A).

Lemma 5.3.15 (Trace sigma-algebra). Let pΩ,Aq be a measurable space and letA Ď Ω be a subset of Ω (which is not necessarily an element of A). Then it holdsthat pA,A \Aq is a measurable space.

The proof of Lemma 5.3.15 is clear and therefore omitted. The next lemma andits proof can, e.g., be found as Corollary 1.83 in Klenke [13].

Lemma 5.3.16 (Trace sigma-algebras and generation of sigma-algebras). Let Ω bea set, let A Ď PpΩq be a subset of the power set of Ω, and let A Ď Ω be a subset ofΩ. Then

A \ σΩpAq “ σApA \Aq . (5.49)

Proof of Lemma 5.3.16. Let ι : A Ñ Ω be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that

ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.50)


This implies that

σApA \Aq “ σAptAXB P PpAq : B P Auq “ σA`

ι´1pBq P PpAq : B P A

(˘

“

ι´1pBq P PpAq : B P σΩpAq

(

“ tAXB P PpAq : B P σΩpAqu “ A \ σΩpAq .(5.51)


Lemma 5.3.17 (Trace topology). Let pE, Eq be a topological space and let A Ď Ebe a subset of A. Then it holds that pA,A \ Eq is a topological space.


Definition 5.3.18 (Generation of topologys). Let E be a set and let E Ď PpEq bea subset of the power set of E. Then we denote by τEpEq the set given by

τEpEq “č

A is a topologyon E with EĎA

A (5.52)

and we call τEpEq the smallest topology on E which contains E.

Lemma 5.3.19 (Topological spaces and continuous mappings). Let E and F be sets,let F Ď PpF q be a subset of the power set of F , and let f : E Ñ F be a mapping.Then

τE`

f´1pAq : A P F

(˘

“

f´1pAq : A P τF pFq

(

. (5.53)

Proof of Lemma 5.3.19. Throughout this proof let E Ď PpEq, E Ď PpEq and F ĎPpF q be the sets given by

E “ τE`

f´1pAq : A P F

(˘

, E “


(

(5.54)

and F “

A P PpF q : f´1pAq P E

(

. (5.55)

Observe that pE, Eq, pE, Eq and pF, Fq are topological spaces and that F Ď F andtf´1pAq : A P Fu Ď E . Hence, we obtain that

τF pFq Ď F and E Ď E . (5.56)

This proves that

E “


(

Ď

f´1pAq : A P F

(

Ď E Ď E . (5.57)

This shows that E “ E . The proof of Lemma 5.3.19 is thus completed.


Lemma 5.3.20 (Trace topologys and generation of topologys). Let E be a set, letE Ď PpEq be a subset of the power set of E, and let A Ď E be a subset of E. Then

A \ τEpEq “ τApA \ Eq . (5.58)

Proof of Lemma 5.3.20. Let ι : A Ñ E be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that

ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.59)

This and Lemma 5.3.19 imply that

τApA \ Eq “ τAptAXB P PpAq : B P Euq “ τA`

ι´1pBq : B P E

(˘

“

ι´1pBq : B P τEpEq

(

“ tAXB P PpAq : B P τEpEqu “ A \ τEpEq .(5.60)


Lemma 5.3.21 (Open balls generate the topologys associated to a distance-typefunction). Let E be a set, let T Ď R be a set, and let d : E ˆ E Ñ T be a functionwith the property that @x, y, z P E : dpx, xq ď 0 and dpx, zq ď dpx, yq ` dpy, zq. Then

τpdq “ τE`

ty P E : dpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

. (5.61)

Lemma 5.3.21 is an immediate consequence from Defintion 2.4.4 and Lemma 2.4.5.In the next result, Corollary 5.3.22, we study the trace set of a topological space as-sociated to a metric.

Corollary 5.3.22 (Traces and metric spaces). Let pE, dEq be a metric space and letA Ď E be a subset of E. Then A \ τpdEq “ τpdE|AÂq.

Proof of Corollary 5.3.22. Lemma 5.3.20 and Lemma 5.3.21 imply that

A \ τpdEq “ A \ τE`

ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

“ τA`

A \

ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

“ τA`

ty P A : dEpx, yq ă εu P PpAq : x P E, ε P p0,8q(˘

“ τA`

ty P A : dEpx, yq ă εu P PpAq : x P A, ε P p0,8q(˘

“ τpdE|AÂq.

(5.62)



Corollary 5.3.23 (Traces and Borel sigma algebras on topological spaces). LetpE, Eq be a topological space and let A Ď E be a subset of E. Then

A \ BpEq “ A \ σEpEq “ σApA \ Eq “ BpAq. (5.63)

Corollary 5.3.23 is an immediate consequence from Lemma 5.3.16. The nextresult, Corollary 5.3.24, specalises Corollary 5.3.23 to the case where the underlyingtopological space is generated by a metric.

Corollary 5.3.24 (Traces and Borel sigma algebras on metric spaces). Let pE, dEqbe a metric space and let A Ď E be a subset of E. Then

A \ BpEq “ A \ σE`

τpdEq˘

“ σA`

τpdE|AÂq˘

“ BpAq. (5.64)

Corollary 5.3.24 is an immediate consequence of Corollary 5.3.23 and Corol-lary 5.3.22.

5.3.1.5 Fourier transform on normed vector spaces

In Lemma 5.3.4 it has been proved that the Fourier transforms of measures on sepa-rable normed vector spaces determine the measures uniquely. The next result, The-orem 5.3.25, provides a generalization to Lemma 5.3.4. The proof of Theorem 5.3.25uses Corollary 5.3.24 above.

Theorem 5.3.25 (Characteristic functional). Let pV, ¨V q be a normed R-vectorspace. Then FV |tµPDpFV q : µ is a.s. separably supportedu is injective.

Proof of Theorem 5.3.25. Let µ1, µ2 P DpFV q be two a.s. separably supported finitemeasures with the property that

FV pµ1q “ FV pµ2q (5.65)

The assumption that µ1 and µ2 are a.s. separably supported ensures that there existseparable and closed sets A1, A2 P BpV q with the property that

µ1pV zA1q “ µ2pV zA2q “ 0. (5.66)

Lemma 5.3.6 implies that the set A1YA2 is separable. This and Lemma 5.3.7 provethat the set

spanpA1 Y A2q (5.67)


is separable. Next let U be the set given by

U “ spanpA1 Y A2q. (5.68)

Lemma 5.3.5 and (5.66) prove that U is separable. The pair pU, ¨V |Uq is thus aclosed and separable R-vector subspace of pV, ¨V q. Moreover, equation (5.66) andthe fact that A1 Ď U and A2 Ď U prove that

µ1pV zUq “ µ2pV zUq “ 0. (5.69)

Next let ϕ P U 1 “ LpU,Rq be arbitrary. Theorem 5.2.1 then implies that there existsa ψ P V 1 “ LpV,Rq with the property that ψ|U “ ϕ. Equation (5.65) hence provesthat

`

FV µ1

˘

pψq “`

FV µ2

˘

pψq. (5.70)

Next note that Corollary 5.3.24 and the fact that U P BpV q imply that BpUq “U \BpV q “ PpUqXBpV q. This and (5.69) ensure that for all k P t1, 2u it holds that

`

FV µk˘

pψq “

ż

V

ei¨ψpxq µkpdxq “

ż

V

1Upxq ei¨ψpxq µkpdxq

“

ż

V

1Upxq ei¨ϕpxq µkpdxq “

ż

U

1Upxq ei¨ϕpxq µk|BpUqpdxq “

`

FU µk|BpUq˘

pϕq.

(5.71)

Combining (5.70) and (5.71) proves that`

FU µ1|BpUq˘

pϕq “`

FU µ2|BpUq˘

pϕq. Asϕ P U 1 was arbitrary, we obtain that

FU`

µ1|BpUq˘

“ FU`

µ2|BpUq˘

. (5.72)

Lemma 5.3.4 and the fact that U is separable hence imply that µ1|BpUq “ µ2|BpUq.This, (5.69) and the fact that BpUq “ U \ BpV q (see Corollary 5.3.24 above) implythat for all A P BpV q it holds that

µ1pAq “ µ1pAX Uq ` µ1pAzUq “ µ1pAX Uq “ µ1|U\BpV qpAX Uq

“ µ1|BpUqpAX Uq “ µ2|BpUqpAX Uq “ µ2pAX Uq

“ µ2pAzUq ` µ2pAX Uq “ µ2pAq.

(5.73)



Corollary 5.3.26 (Independence). Let pΩ,F ,Pq be a probability space, let I be aset, let pVi, ¨Viq, i P I, be normed R-vector spaces, let Xi P MpF ,BpViqq, i P I,satisfy for all i P I that XipPqBpViq is almost surely separably supported, and assumefor all finite sets J Ď I and all φi P CpVi,Rq, i P J , with

ř

iPJ supxPVi |φipxq| ă 8that

E“ś

iPJ φipXiq‰

“ś

iPJ ErφipXiqs . (5.74)

Then it holds that the family Xi, i P I, is P-independent.

Proof of Corollary 5.3.26. Observe that (5.74) ensures that for all finite sets J Ď Iand all φi P CpVi,Cq, i P J , with

ř

iPJ supxPVi |φipxq| ă 8 it holds that

E“ś

iPJ φipXiq‰

“ś

iPJ ErφipXiqs . (5.75)

This implies that for all J P tA Ď I : 0 ă #IpAq ă 8u, ϕ P LpîPJVi,Rq it holdsthat

ˆ

FîPJVi

´

pXiqiPJpPqbiPJBpViq¯

˙

pϕq “ E“

eiϕppXiqiPJ q‰

“ E

«

exp

˜

iϕ

˜

ÿ

iPJ

p1tiupjqXjqjPJ

¸¸ff

“ E

«

exp

˜

iÿ

iPJ

ϕpp1tiupjqXjqjPJq

¸ff

“ E

«

ź

iPJ

exp`

iϕ`

p1tiupjqXjqjPJ˘˘

ff

“ź

iPJ

E“

exp`

iϕ`

p1tiupjqXjqjPJ˘˘‰

“ź

iPJ

ż

Vi

exp`

iϕpp1tiupjqxjqjPJq˘

XipPqBpViqpdxiq.

(5.76)

Hence, we obtain that for all J P tA Ď I : 0 ă #IpAq ă 8u, ϕ P LpîPJVi,Rq itholds that

ˆ

FîPJVi

´

pXiqiPJpPqbiPJBpViq¯

˙

pϕq

“

ż

îPJVi

ź

iPJ

exp`

iϕpp1tiupjqxjqjPJq˘ `

biPJ“

XipPqBpViq‰˘

pdpxiqiPJq

“

ż

îPJVi

exppiϕppxiqiPJqq`

biPJ“

XipPqBpViq‰˘

pdpxiqiPJq

“

ˆ

FîPJVi

´

biPJ“

XipPqBpViq‰

¯

˙

pϕq.

(5.77)


Combining this with Theorem 5.3.25 establishes that

pXiqiPJpPqbiPJBpViq “ biPJ“

XipPqBpViq‰

. (5.78)

This completes the proof of Corollary 5.3.26.

Class exercise 5.3.27. Let pV, ¨V q be a normed R-vector space and let µ : BpV q Ñr0,8s be a finite measure on pV,BpV qq. What is then the characteristic functionalFV pµq?

Class exercise 5.3.28. Let pV, ¨V q be a normed R-vector space and let µ1, µ2 : BpV q Ñr0,8s be two finite measures on pV,BpV qq with FV pµ1q “ FV pµ2q. Provide a condi-tion which is sufficient to ensure that µ1 “ µ2.

5.3.2 Covariances on normed vector spaces

5.3.2.1 Regularities for correlations on normed vector spaces

Proposition 5.3.29 (A boundedness result for correlations on normed vector spaces).Let K P tR,Cu, r P p0,8q, let pV, ¨V q be a normed K-vector space, and letµ : BpV q Ñ r0,8s be a measure with the property that for all ϕ P V 1 it holds thatş

V|ϕpxq|r µpdxq ă 8. Then

supϕPV 1zt0u

„

ş

V|ϕpxq|r µpdxq

ϕrV 1

ă 8. (5.79)

Proof of Proposition 5.3.29. Throughout this proof let Vn Ď V 1, n P N, be the setswhich satisfy for all n P N that

Vn “"

ϕ P V 1 :

ż

V

|ϕpxq|r µpdxq ď n

*

. (5.80)

Fatou’s lemma proves that for all n P N, ψ P V 1, pϕkqkPN Ď Vn with limkÑ8 ϕk ´ ψV 1 “0 it holds that

ż

V

|ψpxq|r µpdxq “

ż

V

ˇ

ˇ

ˇlimkÑ8

ϕkpxqˇ

ˇ

ˇ

r

µpdxq “

ż

V

limkÑ8

|ϕkpxq|r µpdxq

“

ż

V

lim infkÑ8

|ϕkpxq|r µpdxq ď lim inf

kÑ8

ż

V

|ϕkpxq|r µpdxq ď lim inf

kÑ8rns “ n.

(5.81)


This implies that for every n P N it holds that Vn Ď V 1 is a closed subset of V 1. Theassumption that for all ϕ P V 1 it holds that

ş

V|ϕpxq|r µpdxq ă 8 proves that

YnPNVn “ V 1. (5.82)

The fact that Vn Ď V 1, n P N, are closed sets, the fact that pV 1, ¨V 1q is complete(see Lemma 3.5.13) and the Baire category theorem (see Theorem 4.5.2) hence prove

that there exists an N P N such that rVN sc ‰ V 1. Lemma 4.5.1 therefore shows thatthere exist ψ P VN , ε P p0,8q such that

tϕ P V 1 : ϕ´ ψV 1 ď εu Ď VN . (5.83)

This implies that for all ϕ P V 1 with ϕV 1 ď ε it holds that

ż

V

|ϕpxq|r µpdxq “

ż

V

|pϕ` ψqpxq ´ ψpxq|r µpdxq

ď 2r„ż

V

|pϕ` ψqpxq|r µpdxq `

ż

V

|ψpxq|r µpdxq

ď 2r„

N `

ż

V

|ψpxq|r µpdxq

ď 2pr`1qN ă 8.

(5.84)

This, in turn, proves that for all ϕ P V 1zt0u it holds that

ż

V

|ϕpxq|r µpdxq “ϕrV 1

εr

ż

V

ˇ

ˇ

ˇ

ˇ

ε ¨ ϕpxq

ϕV 1

ˇ

ˇ

ˇ

ˇ

r

µpdxq ď2pr`1qN ϕrV 1

εră 8. (5.85)

This implies (5.79). The proof of Proposition 5.3.29 is thus completed.

The next result, Corollary 5.3.30, specialises Proposition 5.3.29 to the case wherer P p0,8q in Proposition 5.3.29 is a natural number.


Corollary 5.3.30 (A continuity result for correlations on normed vector spaces).Let K P tR,Cu, k P N, let pV, ¨V q be a normed K-vector space, and let µ : BpV q Ñr0,8s be a measure with the property that for all ϕ P V 1 it holds that

ş

V|ϕpxq|k µpdxq ă

8. Then

(i) it holds that

supϕ1,...,ϕkPV 1zt0u

„

ş

V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq

ϕ1V 1 ¨ . . . ¨ ϕkV 1

ă 8 (5.86)

and

(ii) it holds that the symmetric k-linear form

V 1 ˆ ¨ ¨ ¨ ˆ V 1 Q pϕ1, . . . , ϕkq ÞÑ

ż

V

ϕ1pxq . . . ϕkpxqµpdxq P K (5.87)

is continuous.

Proof of Corollary 5.3.30. Proposition 5.3.29 implies that

supϕPV 1zt0u

«

ş

V|ϕpxq|k µpdxq

ϕkV 1

ff

ă 8. (5.88)

Holder’s inequality hence shows that

supϕ1,...,ϕkPV 1zt0u

„

ş

V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq

ϕ1V 1 ¨ . . . ¨ ϕkV 1

ď supϕ1,...,ϕkPV 1zt0u

«

kź

l“1

«

ş

V|ϕlpxq|

k µpdxq

ϕlkV 1

ffff1k

“

«

supϕPV 1zt0u

ş

V|ϕpxq|k µpdxq

ϕkV 1

ff1k

ă 8.

(5.89)

This proves (5.86). Inequality (5.86), in turn, establishes (5.87). The proof of Corol-lary 5.3.30 is thus completed.


5.3.2.2 Covariances on normed vector spaces

Definition 5.3.31 (Covariance of a probability measure*). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let µ : BpV q Ñ r0,8s be a probabilitymeasure with the property that for all ϕ P V 1 it holds that

ş

V|ϕpvq|2 µpdvq ă 8.

Then we denote by Covpµq : V 1 ˆ V 1 Ñ K the mapping with the property that for allϕ, ψ P V 1 it holds that

pCov µqpϕ, ψq “`

Covpµq˘

pϕ, ψq

“

ż

V

„

ϕpvq ´

ż

V

ϕpuqµpduq

„

ψpvq ´

ż

V

ψpuqµpduq

µpdvq

“

ż

V

ϕpvqψpvqµpdvq ´

„ż

V

ϕpvqµpdvq

„ż

V

ψpvqµpdvq

(5.90)

and we call Covpµq the covariance of µ.

Definition 5.3.32 (Covariance of a random variable*). Let pΩ,F ,Pq be a probabilityspace, let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and let X : Ω Ñ Vbe an F/BpV q-measurable mapping with the property that for all ϕ P V 1 it holds thatE“

|ϕpXq|2‰

ă 8. Then we denote by CovpXq : V 1 ˆ V 1 Ñ K the mapping given byCovpXq “ CovpXpPqqBpV q.

Lemma 5.3.33 (Properties of the covariance*). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let µ : BpV q Ñ r0,8s be a probability measure with theproperty that for all ϕ P V 1 it holds that

ş

V|ϕpvq|2 µpdvq ă 8. Then Covpµq : V 1 ˆ

V 1 Ñ K is

• nonnegative, i.e., @ϕ P V 1 : pCov µqpϕ, ϕq P r0,8q,

• Hermitian, i.e., @ϕ, ψ P V 1 : pCov µqpϕ, ψq “ pCov µqpψ, ϕq,

• sesquilinear, i.e., @φ, ϕ, ψ P V 1, a P K : pCov µqpaφ` ϕ, ψq “ apCov µqpφ, ψq `pCov µqpϕ, ψq,

• continuous, and

• it holds that

8 ą CovpµqLp2qpV 1,Kq “ supϕ,ψPV 1zt0u

„

|pCov µqpϕ, ψq|

ϕV 1 ψV 1

ď

ż

V

v2V µpdvq. (5.91)

5.4. PROBABILITY MEASURES ON HILBERT SPACES 159

Proof of Lemma 5.3.33*. First of all, observe that the nonnegativity, the Hermitian-ity, and the sesquilinearity of Covpµq follow immediately from Definition 5.3.31. Nextnote that Corollary 5.3.30 ensures that Covpµq is continuous and that

8 ą CovpµqLp2qpV 1,Rq “ supϕ,ψPV 1zt0u

„

|pCov µqpϕ, ψq|

ϕV 1 ψV 1

. (5.92)

Moreover, we observe that Holder’s inequality implies that for all ϕ, ψ P V 1 it holdsthat

|pCov µqpϕ, ψq| ď

ż

V

ˇ

ˇ

ˇ

ˇ

ϕpvq ´

ż

V

ϕpuqµpduq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ψpvq ´

ż

V

ψpuqµpduq

ˇ

ˇ

ˇ

ˇ

µpdvq

ď

«

ż

V

ˇ

ˇ

ˇ

ˇ

ϕpvq ´

ż

V

ϕpuqµpduq

ˇ

ˇ

ˇ

ˇ

2

µpdvq

ff12 «ż

V

ˇ

ˇ

ˇ

ˇ

ψpvq ´

ż

V

ψpuqµpduq

ˇ

ˇ

ˇ

ˇ

2

µpdvq

ff12

ď

„ż

V

|ϕpvq|2 µpdvq

12 „ż

V

|ψpvq|2 µpdvq

12

ď ϕV 1 ψV 1

ż

V

v2V µpdvq.

(5.93)


5.4 Probability measures on Hilbert spaces

5.4.1 Nuclear operators on Hilbert spaces

Below we study square integrable Hilbert space valued random variables. The co-variance operator associated to such a random variable is a nuclear operator. Tostudy such covariance operators, we need a few more properties of nuclear operatorson Hilbert spaces.

Definition 5.4.1 (*). Let K P tR,Cu and let V be a normed K-vector space. Thenwe say that β is an inner product on V (we say that β is a scalar product on V ) ifand only if β PMpV ˆ V,Kq is a mapping from V ˆ V to K which satisfies

(i) that it holds for all x P V zt0u that βpx, xq P p0,8q,

(ii) that it holds for all x, y P V that βpx, yq “ βpy, xq, and

(iii) that it holds for all λ P K, x, y, z P V that βpx, y ` λzq “ βpx, yq ` λβpx, zq.


Lemma 5.4.2 (Completeness of the space of nuclear operators). Let K P tR,Cu andlet pV, ¨V q and pW, ¨W q be K-Banach spaces. Then the pair pL1pV,W q, ¨L1pV,W q

q

is a K-Banach space.

Lemma 5.4.2 is, for example, proved as Theorem VI.5.3 (c) in Werner [23].

5.4.1.1 Rank-1 operators on inner product spaces

In the next step we extend the notion of rank-1 operators (cf. Definition 3.5.18 above)in the case of inner product spaces.

Definition 5.4.3 (Rank-1 operators in inner product spaces*). Let K P tR,Cu,let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-inner product spaces, and let v P V ,w P W . Then we denote by w b v P LpV,W q the mapping given by

pw b vq “ w b pV Q u ÞÑ 〈v, u〉V P Kq “ w b 〈v, ¨〉V . (5.94)

Remark 5.4.4 (Properties of rank-1 operators on inner product spaces*). Let K P

tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, and let v P V ,w P W . Then observe

(i) that for all u P V it holds that

pw b vqpuq “`

w b 〈v, ¨〉V˘

puq “ w 〈v, u〉V , (5.95)

(ii) that

w b v P L1pV,W q Ď L2pV,W q “ HSpV,W q Ď LpV,W q, (5.96)

and

(iii) that

w b vL1pV,W q“ w b vL2pV,W q

“ w b vLpV,W q “ wW vV . (5.97)


5.4.1.2 Traces of nuclear operators

See, e.g., Lemma VI.5.6 in Werner [23] for the next lemma.

Lemma 5.4.5 (Preparatory lemma for the trace of a nuclear operator*). Let K P

tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A P L1pHq, and let pvnqnPN ĎH, pwnqnPN Ď H satisfy for all x P H that

ř8

n“1 vnH wnH ă 8 and

Ax “8ÿ

n“1

pwn b vnqpxq. (5.98)

Then it holds for all orthonormal bases B Ď H of H that

ÿ

bPB

|〈b, Ab〉H | ď8ÿ

n“1

vnH wnH ă 8 and8ÿ

n“1

〈vn, wn〉H “ÿ

bPB

〈b, Ab〉H . (5.99)

Proof of Lemma 5.4.5*. Observe that the Holder inequality proves for all orthonor-mal bases B Ď H of H that

ÿ

bPB

|〈b, Ab〉H | “ÿ

bPB

ˇ

ˇ

ˇ

ˇ

ˇ

⟨b,

8ÿ

n“1

pwn b vnqpbq

⟩H

ˇ

ˇ

ˇ

ˇ

ˇ

ďÿ

bPB

8ÿ

n“1

|〈b, pwn b vnqpbq〉H |

“ÿ

bPB

8ÿ

n“1

|〈b, wn〉H 〈vn, b〉H | “8ÿ

n“1

«

ÿ

bPB

|〈b, wn〉H | |〈b, vn〉H |

ff

ď

8ÿ

n“1

«

ÿ

bPB

|〈b, wn〉H |2

ff12 «

ÿ

bPB

|〈b, vn〉H |2

ff12

“

8ÿ

n“1

vnH wnH ă 8.

(5.100)

Moreover, note for all orthonormal bases B Ď H of H that

8ÿ

n“1

〈vn, wn〉H “8ÿ

n“1

ÿ

bPB

〈b, vn〉H 〈b, wn〉H “8ÿ

n“1

ÿ

bPB

〈b, wn〉H 〈vn, b〉H

“

8ÿ

n“1

ÿ

bPB

〈b, pwn b vnqpbq〉H “ÿ

bPB

⟨b,

8ÿ

n“1

pwn b vnqpbq

⟩H

“ÿ

bPB

〈b, Ab〉H .(5.101)



Lemma 5.4.5 allow us to introduce the concept of the trace of a nuclear operatoron a Hilbert space.

Definition 5.4.6 (Trace of a nuclear operator*). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A P L1pHq. Then we denote by traceHpAq P K the el-ement of K which satisfies that for all orthonormal basis B Ď H of H it holds that

traceHpAq “ÿ

bPB

〈b, Ab〉H P K. (5.102)

5.4.1.3 Absolute value operators

Lemma 5.4.7 (Preparatory lemma for the absolute value operator*). Let K P

tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be K-Hilbert spaces, and let A P

LpH,Uq. Then it holds that A˚A P LpHq is nonnegative and symmetric.

Proof of Lemma 5.4.7. Note that for all v, w P H it holds that

〈v, A˚Aw〉H “⟨rA˚s˚ v, Aw

⟩U“ 〈Av,Aw〉U “ 〈A

˚Av,w〉H . (5.103)

This proves that A˚A is symmetric and that for all v P H it holds that

〈v,A˚Av〉H “ 〈Av,Av〉U “ Av2U ě 0. (5.104)

Hence, we obtain that A˚A is nonnegative. The proof of Lemma 5.4.7 is thus com-pleted.

Lemma 5.4.7 and Definiton 3.5.16 allows us to introduce the absolute value op-erator of a bounded linear operator.

Definition 5.4.8 (The absolute value operator of a bounded linear operator*). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be K-Hilbert spaces, and letA P LpH,Uq. Then we denote by |A| P LpHq the mapping given by

|A| “ rA˚As12P LpHq. (5.105)

Lemma 5.4.9 (The trace of the absolute value operator*). Let K P tR,Cu, letpV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, let A P LpV,W q, and letB Ď V be an orthonormal basis of V . Then

ř

bPB 〈b, |A| b〉V ă 8 if and only ifA P L1pV,W q and in that case it holds that

ÿ

bPB

〈b, |A| b〉V “ traceV p|A|q “ AL1pV,W q. (5.106)


Lemma 5.4.9 can, e.g., be established by using the theory of singular values; see,e.g., Werner [23].

Proposition 5.4.10 (Properties of the absolute value operator of a bounded linearoperator). Let K P tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbertspaces, and let A P LpV,W q. Then

(i) |A| P LpV q is nonnegative and symmetric,

(ii)ˇ

ˇ |A|ˇ

ˇ “ |A|,

(iii) for all v P V it holds that

AvW “ |A|vV , (5.107)

(iv) for all i P t1, 2u it holds that A P LipV,W q if and only if |A| P LipV q,

(v) A P L2pV,W q if and only if |A|2 “ A˚A P L1pV q and it that case it holds that

A2L2pV,W q“›

›|A|2›

›

L1pV q“ traceV p|A|

2q, (5.108)

(vi) A P L1pV,W q if and only if |A|12P L2pV q and it that case it holds that

AL1pV,W q“›

›|A|12›

›

2

L2pV q“ traceV p|A|q, (5.109)

and

(vii) if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A is symmetric and nonnegative,then |A| “ A.

Proof of Proposition 5.4.10. Definition 3.5.16 ensures that |A| is nonnegative and

symmetric. This shows that ||A|| ““

|A|˚ |A|‰12

““

|A|2‰12

“ |A|. Furthermore,observe that for all v P V it holds that

Av2W “ 〈Av,Av〉W “ 〈v, A˚Av〉V “@

v, |A| |A| vD

V

“@

|A| v, |A| vD

V“ |A|v2V .

(5.110)

This proves (5.107). The identity (5.107), in turn, shows that A P L2pV,W q if andonly if |A| P L2pV q. Lemma 5.4.9 implies that A P L1pV,W q if and only if thereexists an orthonormal basis B Ď V of V such that

ÿ

bPB

〈b, |A| b〉V ă 8. (5.111)


Furthermore, Lemma 5.4.9 proves that |A| P L1pV q if and only if there exists anorthonormal basis B Ď V of V such that

ÿ

bPB

〈b, |A| b〉V ă 8. (5.112)

Combining (5.111) and (5.112) proves that A P L1pV,W q if and only if |A| P L1pV q.Next let B Ď V be an orthonormal basis of V and observe that

ÿ

bPB

Ab2W “ÿ

bPB

〈Ab,Ab〉V “ÿ

bPB

〈b, A˚Ab〉V “ÿ

bPB

⟨b, |A|2 b

⟩V. (5.113)

This and Lemma 5.4.9 prove Item (v). Item (iv), Item (v) and Lemma 5.4.9 proveItem (vi). Moreover, note that if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A issymmetric and nonnegative, then

rA˚As12““

A2‰12“ A. (5.114)


Theorem 5.4.11 (Spectral decomposition for compact operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P KpH,Hq be a symmetric com-pact operator. Then A is a diagonal linear operator.

Theorem 5.4.11 is, e.g., proved as Theorem VI.3.2 in Werner [23].

5.4.2 Covariances on Hilbert spaces

In this subsection we intend to extend the notion of the covariance (see Defini-tion 5.3.31 above) in the case of Hilbert spaces.

Class exercise 5.4.12 (*). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and letµ : BpHq Ñ r0,8s be a probability measure which satisfies for all w P H thatş

H|〈w, v〉H |

2 µpdvq ă 8. Does there exists a C P LpHq such that for all v, w P H itholds that

〈v, Cw〉H “ pCov µq`

H Q u ÞÑ 〈v, u〉H P R, H Q u ÞÑ 〈w, u〉H P R˘

. (5.115)

Definition 5.4.13 (Covariance operator of a probability measure on a Hilbertspace*). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be aprobability measure which satisfies for all w P H that

ş

H|〈w, v〉H |

2 µpdvq ă 8. Thenwe denote by CovOppµq P LpHq the unique bounded linear operator which satisfiesfor all v, w P H that

〈v,CovOppµqw〉H “ pCov µq`

H Q u ÞÑ 〈v, u〉H P R, H Q u ÞÑ 〈w, u〉H P R˘

. (5.116)


Lemma 5.4.14 (Properites of covariance operators of probability measures with fi-nite second moments*). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñr0,8s be a probability measure with

ş

Hv2H µpdvq ă 8. Then

(i) it holds that CovOppµq is symmetric and nonnegative,

(ii) it holds that CovOppµq is a nuclear operator, and

(iii) it holds that

traceHpCovOppµqq “ CovOppµqL1pHqď

ż

H

v2H µpdvq P r0,8q. (5.117)

Proof of Lemma 5.4.14*. Throughout this proof assume w.l.o.g. that B ‰ H. Non-negativity and symmetry of CovOppµq follows immediately from Lemma 5.3.33. Nextobserve that for all orthonormal bases B Ď H of H it holds thatÿ

bPB

〈b, |CovOppµq| b〉H “ÿ

bPB

〈b,CovOppµqb〉H

“ÿ

bPB

«

ż

H

|〈b, v〉H |2 µpdvq ´

ˇ

ˇ

ˇ

ˇ

ż

H

〈b, v〉H µpdvqˇ

ˇ

ˇ

ˇ

2ff

ďÿ

bPB

ż

H

|〈b, v〉H |2 µpdvq “ sup

BĎBfinite

ÿ

bPB

ż

H

|〈b, v〉H |2 µpdvq

“ supBĎBfinite

ż

H

ÿ

bPB

|〈b, v〉H |2 µpdvq ď sup

BĎBfinite

ż

H

v2H µpdvq

“

ż

H

v2H µpdvq ă 8.

(5.118)

Lemma 5.4.9 hence completes the proof of Lemma 5.4.14.

Definition 5.4.15 (Covariance operator of a Hilbert space valued random variable*).Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, andlet X : Ω Ñ H be an F/BpHq-measurable mapping which satisifes for all v P H thatE“

|〈v,X〉H |2‰

ă 8. Then we denote by CovOppXq P LpHq the linear operator givenby CovOppXq “ CovOp

`

XpPqBpHq˘

.

Note, in the setting of Definition 5.4.13, that for all v, w P H it holds that

〈v,CovOppXqw〉H “ E”

`

〈v,X〉HÉr〈v,X〉Hs˘`

〈w,X〉HÉr〈w,X〉Hs˘

ı

. (5.119)


Definition 5.4.16 (*). Let pΩ,F ,Pq be a probability space, let pV, ¨V q be a Banachspace, and let X P L1pP; ¨V q. Then we denote by ErXs P V the element from Vgiven by

ErXs “

ż

Ω

XpωqPpdωq. (5.120)

Note that the integral appearing on the right hand side of (5.120) is a Bochnerintegral; see Section 3.7 for details.

Proposition 5.4.17 (Properties of the covariance operator of a square integrableHilbert space valued random variable). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hqbe an R-Hilbert space, and let X P L2pP; ¨Hq. Then

(i) it holds that CovOppXq is a symmetric and nonnegative nuclear operator,

(ii) it holds that

CovOppXq “ E“`

X ÉrXs˘

b`

X ÉrXs˘‰

P L1pHq, (5.121)

and

(iii) it holds that

traceHpCovOppXqq “ E“

X ÉrXs2H‰

“ CovOppXqL1pHqP r0,8q.

(5.122)


5.4.3 Karhunen-Loeve expansion

Theorem 5.4.18 (Karhunen-Loeve expansion). Let pΩ,F ,Pq be a probability space,let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let X P L2pP; ¨Hq, let B Ď Hbe an orthonormal basis of H, and let λ : BÑ r0,8q be a globally bounded functionsuch that for all v P H it holds that CovOppXq v “

ř


(i) the random variables p〈b,X ÉrXs〉HqbPB are centered and pairwise uncorre-lated,

(ii) it holds for all b P B that Varp〈b,X ÉrXs〉Hq “ Varp〈b,X〉Hq “ λb,

(iii) it holds that

X “ ErXs `ÿ

bPB

〈b,X ÉrXs〉H b, (5.123)

and

(iv) it holds for all B Ď B that

›

›

›

›

›

X ´

«

ErXs `ÿ

bPB

〈b,X ÉrXs〉H b

ff›

›

›

›

›

L2pP;¨Hq

“

d

ÿ

bPBzB

λb ă 8. (5.124)

Proof of Theorem 5.4.18. Equation (5.123) follows immediately from the fact thatB is an orthonormal basis of H. Furthermore, note that the random variablesp〈b,X ÉrXs〉HqbPB are centered. Next observe that for all b1, b2 P B it holds that

Er〈b1, X ÉrXs〉H 〈b2, X ÉrXs〉Hs “ 〈b1,CovOppXqb2〉H“

ÿ

bPB

〈b1, λb 〈b, b2〉H b〉H

“ λb1 〈b1, b2〉H .

(5.125)

This implies that the random variables p〈b,X ÉrXs〉HqbPB are pairwise uncorre-lated and it shows that for all b P B it holds that

Varp〈b,X〉Hq “ Varp〈b,X ÉrXs〉Hq “ λb. (5.126)


Combining this with (5.123) proves that for all B Ď B it holds that

›

›

›

›

›

X ´

«

ErXs `ÿ

bPB

〈b,X ÉrXs〉H b

ff›

›

›

›

›

L2pP;¨Hq

“

›

›

›

›

›

›

ÿ

bPBzB

〈b,X ÉrXs〉H b

›

›

›

›

›

›

L2pP;¨Hq

“

d

ÿ

bPBzB

E“

|〈b,X ÉrXs〉H |2‰

“

d

ÿ

bPBzB

λb.

(5.127)


5.5 Gaussian measures

5.5.1 Gaussian measures on normed vector spaces

Definition 5.5.1 (One-dimensional Gaussian measures*). We say that µ is a one-dimensional Gaussian measure if and only if µ P MpBpRq, r0,8sq is a measure onpR,BpRqq which satisfies that there exist a, b P R such that for all B P BpRq it holdsthat

µpBq “

ż

txPR : ax`bPBu

1?

2πexp

ˆ

ý2

2

˙

dy. (5.128)

Definition 5.5.2 (Gaussian measures on possibly infinite dimensional spaces*). LetpV, ¨V q be a normed R-vector space. Then µ is called a Gaussian measure onpV, ¨V q (µ is called Gaussian measure) if and only if µ P MpBpV q, r0,8sq is ameasure on pV,BpV qq which satisfies that for all ϕ P V 1 it holds that ϕpµqBpRq is aone-dimensional Gaussian measure.

Definition 5.5.3 (Gaussian distributed random variables*). Let pΩ,F ,Pq be a prob-ability space and let pV, ¨V q be a normed R-vector space. Then we say that X isP-Gaussian distributed on pV, ¨V q (we say that X is Gaussian distributed) if andonly if (it holds that X P MpF ,BpV qq and it holds that XpPqBpV q is a Gaussianmeasure on pV, ¨V q).

5.5. GAUSSIAN MEASURES 169

Example 5.5.4. Let T P p0,8q, m P N, let pΩ,F ,Pq be a probability space, letW : r0, T s ˆ Ω Ñ Rm be a standard Brownian motion with continous sample paths,and let W : Ω Ñ Cpr0, T s,Rmq be the mapping with the property that for all ω P Ω,t P r0, T s it holds that

`

W pωq˘

ptq “ Wtpωq. (5.129)

Then W pPqBpCpr0,T s,Rmqq is a Gaussian measure on`

Cpr0, T s,Rmq, ¨Cpr0,T s,Rmq˘

. To

see this let ϕ P Cpr0, T s,Rmq1 be arbitrary and let PN : Cpr0, T s,Rmq Ñ Cpr0, T s,Rmq,N P N, pN : Cpr0, T s,Rmq Ñ pRmqN`1, N P N, and ιN : pRmqN`1 Ñ Cpr0, T s,Rmq,N P N, be the mappings with the property that for all v P Cpr0, T s,Rmq, N P N,

pa0, a1, . . . , aNq P pRmqN`1, n P t0, 1, . . . , N ´ 1u, t P rnT

N, pn`1qT

Ns it holds that

`

PNpvq˘

ptq ““

n` 1´ tNT

‰

vpnTNq `

“

tNT´ n

‰

vp pn`1qTN

q, (5.130)

`

ιNpa0, a1, . . . , aNq˘

ptq ““

n` 1´ tNT

‰

an `“

tNT´ n

‰

an`1, (5.131)

pNpvq “`

vp0q, vp TNq, vp2T

Nq, . . . , vpT q

˘

. (5.132)

Observe that for all N P N it holds that PN , pN , and ιN are continuous and thatPN “ ιN ˝ pN . Next let ϕN : pRmqN`1 Ñ R, N P N, be the mappings with theproperty that for all N P N it holds that ϕN “ ϕ ˝ ιN . Moreover, note that for everyN P N it holds that

`

ϕ ˝ PN ˝ W˘

pPqBpRq “`

ϕ ˝ PN˘`

W pPq˘

BpRq “`

ϕN ˝ pN ˝ W˘

pPqBpRq (5.133)

is a Gaussian measure. This shows that for all y P R, N P N it holds that

E

”

exp´

i ¨ y ¨ ϕNppNpW qq¯ı

“ exp

ˆ

i ¨ y Ë”

ϕNppNpW qqı

ý2

2¨ Var

´

ϕNppNpW qq¯

˙

“ exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

. (5.134)

Observe that for all N P N, ω P Ω it holds that limNÑ8 ϕN`

pNpW pωqq˘

“ ϕpW pωqq.This, Lebesgue’s theorem of dominated convergence, and (5.134) imply that for ally P R it holds that

E

”

exp´

i ¨ y ¨ ϕpW q¯ı

“ limNÑ8

E

”

exp´

i ¨ y ¨ ϕNppNpW qq¯ı

“ limNÑ8

exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

“ exp

ˆ

ý2

2¨ limNÑ8

E

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

“ exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpW qˇ

ˇ

2ı

˙

. (5.135)

This proves that ϕpW pP qqBpRq is a one-dimensional Gaussian measure and this shows

that W pPqBpCpr0,T s,Rmqq is indeed a Gaussian measure.


5.5.1.1 Fourier transform of a Gaussian measure

Proposition 5.5.5 (Fourier transform of a Gaussian measure*). Let pV, ¨V q be anormed R-vector space and let µ : BpV q Ñ r0,8s be a finite measure. Then µ isGaussian if and only if for all ϕ P V 1 it holds that

ş

V|ϕpvq|2 µpdvq ă 8 and

pFV µqpϕq “ exp

ˆ

i ∫Vϕpvqµpdvq ´ 1

2pCov µqpϕ, ϕq

˙

. (5.136)

Proof of Proposition 5.5.5*. First of all, observe that if µ is a Gaussian measure,then it holds for all ϕ P V 1, ξ P R that

ş


pFV µqpξ ¨ ϕq “

ż

V

ei¨ϕpvq¨ξ µpdvq “

ż

R

ei¨x¨ξ`

ϕpµq˘

pdxq

“ exp

ˆ

i ξ ∫Vϕpvqµpdvq ´ ξ2

2pCov µqpϕ, ϕq

˙

.

(5.137)

This proves the “ñ” direction in the statement of Proposition 5.5.5. It thus remainsto prove the “ð” direction in the statement of Proposition 5.5.5. To this end weassume in the following that for all ϕ P V 1 it holds that

ş


pFV µqpϕq “ exp

ˆ

i ∫Vϕpvqµpdvq ´ 1

2pCov µqpϕ, ϕq

˙

. (5.138)

This implies that for all ϕ P V 1, ξ P R it holds that

pFV µqpξ ¨ ϕq “ exp

ˆ

i ξ ∫Vϕpvqµpdvq ´ ξ2

2pCov µqpϕ, ϕq

˙

. (5.139)

This, in turn, proves that for all ϕ P V 1 it holds that ϕpµq is a Gaussian measure onpR,BpRqq. The proof of Proposition 5.5.5 is thus completed.

5.5.1.2 Covariance of a Gaussian measure

Corollary 5.5.6 (Covariance of Gaussian measures). Let pV, ¨V q be a separablenormed R-vector space and let µk : BpV q Ñ r0,8s, k P t1, 2u, be Gaussian mea-sures which satisfy for all ϕ P V 1 that Covpµ1q “ Covpµ2q and

ş

Vϕpvqµ1pdvq “

ş

Vϕpvqµ2pdvq. Then µ1 “ µ2.

Corollary 5.5.6 is an immediate consequence from Proposition 5.5.5 and fromLemma 5.3.4.


5.5.2 Gaussian measures on Hilbert spaces

Lemma 5.5.7. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, and letµ : BpHq Ñ r0,8s be a measure. Then µ is Gaussian if and only if for all v P H itholds that Rep〈v, µ〉Hq is a Gaussian measure.


5.5.2.1 Karhunen-Loeve expansion

Corollary 5.5.8 (Karhunen-Loeve expansion for Gaussian distributed random vari-ables). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let X P L2pP; ¨Hq be Gaussian distributed, let B Ď H be an orthonor-mal basis of H, and let λ : B Ñ r0,8q be a globally bounded function such that forall v P H it holds that CovOppXq v “

ř

bPB λb 〈b, v〉H b. Then the random variables1?λb〈b,X ÉrXs〉H , b P λ´1pp0,8qq, are independent identically distributed (i.i.d.)

standard normal random variables and it holds P-a.s. that

X “ ErXs `ÿ

bPλ´1pp0,8qq

a

λb

„

〈b,X ÉrXs〉H?λb

b. (5.140)

Proof of Corollary 5.5.8. First of all, observe that Theorem 5.4.18 together withthe assumption that XpPq is a Gaussian measure imply that the random variables

1?λb〈b,X ÉrXs〉H , b P λ´1pp0,8qq, are i.i.d. standard normal random variables.

Moreover, Theorem 5.4.18 ensures that for all b P λ´1pt0uq it holds that

E“

|〈b,X ÉrXs〉H |2‰

“ Varp〈b,X ÉrXs〉Hq “ λb “ 0. (5.141)

This proves that for all b P λ´1pt0uq it holds P-a.s. that

〈b,X ÉrXs〉H “ 0. (5.142)

This shows that it holds P-a.s. that

X ÉrXs “ÿ

bPλ´1pp0,8qq

〈b,X ÉrXs〉H b. (5.143)

Equation (5.143) implies (5.140). The proof of Corollary 5.5.8 is thus completed.


5.5.2.2 Fourier transform of a Gaussian measure

The next result, Corollary 5.5.9, is a direct consequence of Proposition 5.5.5 above.

Corollary 5.5.9 (Fourier transform of a Gaussian measure on a Hilbert space*).Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be a prob-ability measure. Then µ is Gaussian if and only if for all v P H it holds thatş

H|〈v, w〉H |

2 µpdwq ă 8 and

pFV µqp〈v, ¨〉Hq “ exp`

i ∫H 〈v, w〉H µpdwq ´12〈v,CovOppµq v〉H

˘

. (5.144)

5.5.2.3 Construction of Gaussian measures on Hilbert spaces

In Theorem 5.5.11 below we establish the existence of Gaussian measures on Hilbertspaces. In the proof of Theorem 5.5.11 we use the fact that the set of Cauchysequences of a sequence of strongly measurable mappings is a measurable set; seeLemma 5.5.10 below.

Lemma 5.5.10 (Cauchy sequence). Let pΩ,Fq be a measurable space, let pE, dEqbe a metric space, and let Xn : Ω Ñ E, n P N, be strongly F/pE, dEq-measurablemappings. Then it holds that

ω P Ω: pXnpωqqnPN is a Cauchy-sequence(

P F . (5.145)

Proof of Lemma 5.5.10. First of all, note that the assumption that Xn, n P N,are strongly F/pE, dEq-measurable ensures that for all n,m P N it holds thatpXn, Xmq : Ω Ñ E ˆ E is F/BpE ˆ Eq-measurable. The continuity of the map-ping dE : E ˆ E Ñ r0,8q hence implies that for all n,m P N it holds that thefunction

Ω Q ω ÞÑ dEpXnpωq, Xmpωqq P r0,8q (5.146)

is F/Bpr0,8qq-measurable. This ensures that for all k, n,m P N it holds that

tdEpXn, Xmq ă 1ku P F . (5.147)


This implies that

ω P Ω: pXnpωqqnPN is a Cauchy-sequence(

“

ω P Ω: @ ε P p0,8q : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă ε(

“

ω P Ω: @ k P N : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă 1k(

“ XkPN YNPN Xn,mPNXrN,8q tω P Ω: dEpXnpωq, Xmpωqq ă 1ku

“ XkPN YNPN Xn,mPtN,N`1,... u tdEpXn, Xmq ă 1kulooooooooooomooooooooooon

PF

P F .

(5.148)

This completes the proof of Lemma 5.5.10.

The next result, Theorem 5.5.11, establishes the existence of a Gaussian measureon a Hilbert space with a given mean vector and a given nuclear covariance operator.

Theorem 5.5.11 (Existence of Gaussian measures). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let v P H, and let Q P L1pHq be a nonnegative and symmetric nuclearoperator. Then there exists a Gaussian measure Nv,Q : BpHq Ñ r0,8s which satisfiesfor all w P H that

CovOppNv,Qq “ Q and 〈w, v〉H “ż

H

〈w, x〉H Nv,Qpdxq. (5.149)

Proof of Theorem 5.5.11. Theorem 5.4.11 proves that there exists an orthonormalbasis B Ď H of H and a globally bounded function λ : B Ñ r0,8q such that for allx P H it holds that

Qx “ÿ

bPB

λb 〈b, x〉H b. (5.150)

Proposition 3.6.6 proves thatř

bPB |λb| ă 8. This and Lemma 3.1.13 show thatthe set λ´1pp0,8qq is at most countable. W.l.o.g. we assume that λ´1pp0,8qq iscountable. Hence, there exist a sequence pbnqnPN Ď λ´1pp0,8q with the propertythat for all n,m P N with n ‰ m it holds that bn ‰ bm and with the property that

tbn P B : n P Nu “ λ´1pp0,8qq. (5.151)

Next let pΩ,F ,Pq be a probability space and let Yn : Ω Ñ R, n P N, be i.i.d. standardnormal random variables. Observe that such a probability space does indeed exist.In the next step let XN : Ω Ñ H, N P N, be the mappings with the property thatfor all N P N it holds that

XN “

Nÿ

n“1

a

λbnYnbn. (5.152)


Note that for all N P N it holds that XN is strongly F/pH, ¨Hq-measurable.Lemma 5.5.10 and the completeness of pH, ¨Hq hence prove that the set

A “ tω P Ω: pXNpωqqNPN Ď H is convergent u (5.153)

is in F . This shows that for all N P N it holds that

Ω Q ω ÞÑ 1Apωq ¨XNpωq P H (5.154)

is strongly F/pH, ¨Hq-measurable. Moreover, observe that for all p P r1,8q it holdsthat

›

›

›

›

supNPN

XN2H

›

›

›

›

LppP;|¨|q

“

›

›

›

›

›

supNPN

«

Nÿ

n“1

λbn |Yn|2

ff›

›

›

›

›

LppP;|¨|q

“

›

›

›

›

›

limNÑ8

«

Nÿ

n“1

λbn |Yn|2

ff›

›

›

›

›

LppP;|¨|q

“ limNÑ8

›

›

›

›

›

Nÿ

n“1

λbn |Yn|2

›

›

›

›

›

LppP;|¨|q

ď limNÑ8

Nÿ

n“1

λbn›

›|Yn|2›

›

LppP;|¨|q“›

›|Y1|2›

›

LppP;|¨|q

8ÿ

n“1

λbn ă 8.

(5.155)

In the next step let X : Ω Ñ H be the mapping with the property that for all ω P Ωit holds that

Xpωq “ v ` 1Apωq”

limNÑ8

XNpωqı

“ v ` limNÑ8

r1Apωq ¨XNpωqs . (5.156)

Combining (5.154) and Theorem 2.3.10 proves thatX is strongly F/pH, ¨Hq-measurable.Furthermore, note that for all N P N it holds that

E

«

supM1,M2PtN,N`1,... u

XM1 ´XM22H

ff

“ E

«

8ÿ

n“N`1

›

›

›

a

λbnYnbn

›

›

›

2

H

ff

“

8ÿ

n“N`1

E

„

›

›

›

a

λbnYnbn

›

›

›

2

H

“

8ÿ

n“N`1

λbn E“

|Yn|2H

‰

“

8ÿ

n“N`1

λbn ă 8.

(5.157)

This implies that PpAq “ 1. Moreover, (5.155) ensures that for all p P p0,8q it holdsthat

X ´ vHLppP;|¨|q “

›

›

›

›

›

›

›limNÑ8

p1AXNq

›

›

›

H

›

›

›

›

LppP;|¨|q

“

›

›

›limNÑ8

p1A XNHq

›

›

›

LppP;|¨|qď

›

›

›

›

supNPN

XNH

›

›

›

›

LppP;|¨|q

ă 8.

(5.158)


This, in particular, implies that for all p P p0,8q it holds that

E

„

XpH ` supNPN

XNpH

ă 8. (5.159)

Uniform integrability and the fact that PpAq “ 1 hence prove that

E“

X‰

“ E

”

v ` limNÑ8

p1AXNq

ı

“ v È”

limNÑ8

p1AXNq

ı

“ v ` limNÑ8

Er1AXN s “ v ` limNÑ8

ErXN s

“ v ` limNÑ8

«

Nÿ

n“1

a

λbn ErYns bn

ff

“ v.

(5.160)

In addition, uniform integrability, the fact that PpAq “ 1, and (5.159) ensure thatfor all x, y P H it holds that

〈x,CovOppXqy〉H “ 〈x,CovOppX ´ vqy〉H “ Er〈x,X ´ v〉H 〈y,X ´ v〉Hs“ lim

NÑ8Er〈x,1AXN〉H 〈y,1AXN〉Hs “ lim

NÑ8Er〈x,XN〉H 〈y,XN〉Hs

“ limNÑ8

〈x,CovOppXNqy〉H “ limNÑ8

«

Nÿ

n“1

λbn 〈x, bn〉H 〈y, bn〉H E“

|Yn|2‰

ff

“ limNÑ8

«

Nÿ

n“1

λbn 〈x, bn〉H 〈y, bn〉H

ff

“ limNÑ8

⟨x,

Nÿ

n“1

λbn 〈bn, y〉H bn

⟩H

“ 〈x,Qy〉H .

(5.161)

This implies thatCovOppXq “ Q. (5.162)

Next note that Lebesgue’s theorem of dominated convergence and the fact thatPpAq “ 1 imply that for all w P H it holds that

`

FHXpPq˘

p〈w, ¨〉Hq “ E“

ei〈w,X〉H‰

“ ei〈w,v〉H E“

ei〈w,X´v〉H‰

“ ei〈w,v〉H limNÑ8

E“

ei〈w,1AXN 〉H‰


E“

ei〈w,XN 〉H‰


e´12〈w,CovOppXN qw〉H “ ei〈w,v〉H´

12〈w,CovOppXqw〉H .

(5.163)

This, (5.159), and Corollary 5.5.9 imply thatXpPq is a Gaussian measure. Combiningthis with (5.160) and (5.162) establishes the existence of a probability measure XpPqwith the desired properties. The proof of Theorem 5.5.11 is thus completed.


Corollary 5.5.12 (Existence and uniqueness of Gaussian measures on separableHilbert spaces*). Let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let v P H, andlet Q P L1pHq be a nonnegative and symmetric nuclear operator. Then there existsa unique Gaussian measure Nv,Q : BpHq Ñ r0,8s which satisfies for all w P H that

CovOppNv,Qq “ Q and 〈w, v〉H “ż

H

〈w, x〉H Nv,Qpdxq. (5.164)

Corollary 5.5.12 is an immediate consequence of Theorem 5.5.11 and of Corol-lary 5.5.6.

5.5.2.4 Class exercise on Gaussian distributed random variables

Class exercise 5.5.13 (*). Proposition 5.5.15 and Corollary 5.5.17 contradict toeach other and, in particular, Proposition 5.5.15 or Corollary 5.5.17 are wrong. Re-veal the mistake in the proofs of Proposition 5.5.15 and Corollary 5.5.17 respectively.

Definition 5.5.14 (*). Let pΩ,F ,Pq be a probability space. Then X is called a P-standard normal random variable if and only if it holds that pX PMpF ,BpRqq andXpP qBpRq “ N0,IdRq.

Proposition 5.5.15 (*). Let pΩ,F ,Pq be a probability space, let n P N, and letX1 : Ω Ñ R, . . . , Xn : Ω Ñ R be P-standard normal random variables. Then it holdsthat Ω Q ω ÞÑ pX1pωq, . . . , Xnpωqq P R

n is P-Gaussian distributed on pRn, ¨Rnq.

Proof of Proposition 5.5.15*. Observe that for all ξ “ pξ1, . . . , ξnq P Rn it holds that

E“

ei〈ξ,pX1,...,Xnq〉Rn‰

“

ż

Ω

ei〈ξ,pX1pωq,...,Xnpωqq〉Rn Ppdωq

“

ż

R

. . .

ż

R

ei〈ξ,px1,...,xnq〉Rn X1pPqBpRqpdx1q . . . XnpPqBpRqpdxnq

“

ż

R

. . .

ż

R

«

nź

k“1

eiξkxk

ff

X1pPqBpRqpdx1q . . . XnpPqBpRqpdxnq

“

nź

k“1

„ż

R

eiξkxkXkpPqBpRqpdxkq

“

nź

k“1

„

1?

2π

ż

R

eiξkxk exp´

´pxkq

2

2

¯

dxk

.

(5.165)


Corollary 5.5.9 hence shows for all ξ “ pξ1, . . . , ξnq P Rn that

E“

ei〈ξ,pX1,...,Xnq〉Rn‰

“

nź

k“1

„

1?

2π

ż

R

eiξkx exp´

´x2

2

¯

dx

“

nź

k“1

”

exp´

´pξkq

2

2

¯ı

“ exp´

´ξ2Rn

2

¯

.

(5.166)

This and again Corollary 5.5.9 complete the proof of Proposition 5.5.15.

Proposition 5.5.16 (*). Let pΩ,F ,Pq be a probability space, let X : Ω Ñ R be aP-standard normal random variable, let Z : Ω Ñ R be an

`

12δR´1|BpRq `

12δR1 |BpRq

˘

-distributed random variable with the property that X and Z are independent, and letY : Ω Ñ R be given by Y “ ZX. Then

(i) it holds that X and Y are P-standard normal random variables,

(ii) it holds that X and Y are uncorrelated, i.e., E“

XY‰

“ 0, and

(iii) it does not hold that Ω Q ω ÞÑ pXpωq, Y pωqq P R2 is P-Gaussian distributed.

Proof of Proposition 5.5.16*. First of all, observe that the definition of Y and theassumption that X and Z are independent ensures that for all x P R it holds that

PpY ď xq “ PptY ď xu X tZ “ 1uq ` PptY ď xu X tZ “ ´1uq

“ PptX ď xu X tZ “ 1uq ` Ppt´X ď xu X tZ “ ´1uq

“ 12¨ PpX ď xq ` 1

2¨ Pp´X ď xq .

(5.167)

Next we note that the assumption that XpPqBpRq “ N0,IR ensures that XpPqBpRq “p´XqpPqBpRq “ N0,IR . This and (5.167) imply that for all x P R it holds that

PpY ď xq “ 12¨N0,IRpp´8, xsq `

12¨N0,IRpp´8, xsq “ N0,IRpp´8, xsq. (5.168)

Moreover, we observe that

E“

XY‰

“ E“

XpZXq‰

“ E“

X2Z‰

“ E“

X2‰

E“

Z‰

“ 1 ¨ 0 “ 0. (5.169)

Furthermore, we note that

PpX ` Y “ 0q “ PpX ` ZX “ 0q “ P`

p1` ZqX “ 0˘

“ Pp1` Z “ 0q “ PpZ “ ´1q “ 12.(5.170)


This proves thatΩ Q ω ÞÑ Xpωq ` Y pωq P R (5.171)

is not P-Gaussian distributed. This establishes that

Ω Q ω ÞÑ pXpωq, Y pωqq P R2 (5.172)

is not Gaussian distributed. The proof of Proposition 5.5.16 is thus completed.

The next result, Corollary 5.5.17 below, is an immediate consequence of Propo-sition 5.5.16 above.

Corollary 5.5.17 (*). There exists a probability space pΩ,F ,Pq and P-standard nor-mal random variables X : Ω Ñ R and Y : Ω Ñ R such that Ω Q ω ÞÑ pXpωq, Y pωqq PR2 is not Gaussian distributed on pR2, ¨

R2q.

5.5.2.5 Karhunen-Loeve expansion for Brownian motion

Exercise 5.5.18 (An ordinary differential equation of second order). Let a P Rzt0u,T P p0,8q and let v : r0, T s Ñ R be a twice continuously differentiable function withthe property that for all t P r0, T s it holds that v2ptq “ a vptq.

(i) Prove that for all t P r0, T s it holds that

˜

vptq

v1ptq

¸

“

˜

1 1?a ´

?a

¸˜

et?a 0

0 e´t?a

¸˜

1 1?a ´

?a

¸´1 ˜

vp0q

v1p0q

¸

.

(5.173)

(ii) Prove that there exist A,B P C such that for all t P r0, T s it holds that vptq “Ae

?at `Be´

?at.

Theorem 5.5.19 (Karhunen-Loeve expansion for Brownian motion). Let pΩ,F ,Pqbe a probability space, let T P p0,8q, let W : r0, T s ˆΩ Ñ R be a standard Brownianmotion with continuous sample paths, let W : Ω Ñ L2pBorelp0,T q; |¨|Rq be the mappingwith the property that for all ω P Ω and Borelp0,T q-almost all t P r0, T s it holds that

pW pωqqptq “ Wtpωq, and let ek P L2pBorelp0,T q; |¨|Rq, k P N, be the vectors withthe property that for all k P N and Borelp0,T q-almost all t P p0, T q it holds that

ekptq “?

2?T

sin`

pk ´ 12qπtT

˘

. Then

(i) W is a Gaussian distributed random variable,


(ii) it holds for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q that

`

CovOppW q v˘

ptq “

ż T

0

mintt, su vpsq ds “

ż t

0

ż T

s

vpuq du ds

“

ż T

0

ErWtWss vpsq ds,

(5.174)

(iii) it holds that

σP`

CovOppW q˘

“

"

T 2

r12πs2,

T 2

r32πs2,

T 2

r52πs2, . . .

*

“

"

T 2

π2 pk ´ 12q2 P p0,8q : k P N

*

,

(5.175)

(iv) the set tek : k P Nu is an orthonormal basis of L2pBorelp0,T q; |¨|Rq,

(v) it holds for all v P L2pBorelp0,T q; |¨|Rq that

CovOppW q v “8ÿ

n“1

T 2

π2 rn´ 12s2 〈en, v〉L2pBorelp0,T q;|¨|Rq

en, (5.176)

(vi) the random variables πTrk ´ 12s xek, W yL2pBorelp0,T q;|¨|Rq

, k P N, are i.i.d. standardnormal random variables, and

(vii) it holds that

W “

8ÿ

k“1

Tπ rk´12s

”

π rk´12s

Txek, W yL2pBorelp0,T q;|¨|Rq

ı

ek. (5.177)

Proof of Theorem 5.5.19. First of all, note that for all v P Cpr0, T s,Rq it holds that

ż T

0

vpsqWs ds “ limNÑ8

«

N´1ÿ

n“0

vpnTNqWnT

N

TN

ff

. (5.178)

This proves that for all v P Cpr0, T s,Rq it holds thatşT

0vpsqWs ds is Gaussian

distributed. This and the fact that Cpr0, T s,Rq is dense in L2pBorelp0,T q; |¨|Rq implies

that for all v P L2pBorelp0,T q; |¨|Rq it holds thatşT

0vpsqWs ds is Gaussian distributed.


It thus holds that W is a Gaussian distributed random variable. Next observe thatfor all v, w P L2pBorelp0,T q; |¨|Rq it holds that

xw,CovOppW qvyL2pBorelp0,T q;|¨|Rq“ E

„ż T

0

vpsqWs ds

ż T

0

wpuqWu du

“

ż T

0

wpuq

ż T

0

ErWsWus vpsq ds du “

ż T

0

wpuq

ż T

0

minps, uq vpsq ds du.

(5.179)

This and Fubini’s theorem show that for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q it holds that

`

CovOppW q v˘

ptq “

ż T

0

ErWsWts vpsq ds “

ż T

0

mints, tu vpsq ds

“

ż T

0

ż mintu,tu

0

vpuq ds du “

ż t

0

ż T

0

1tsďuu vpuq du ds

“

ż t

0

ż T

s

vpuq du ds.

(5.180)

This proves (ii). In the next step let µ P R and let v : p0, T q Ñ R be an Bpp0, T qq/BpRq-measurable function with the property that

şT

0|vpsq|2 ds “ 1 and with the property

thatµ ¨ v “ CovOppW q v. (5.181)

Equation (5.181) implies that for Borelr0,T s-almost all t P p0, T q it holds that

µ ¨ vptq “

ż T

0

minps, tq vpsq ds. (5.182)

Next let w : r0, T s Ñ R be the function with the property that for all t P r0, T s itholds that

wptq “

ż t

0

ż T

s

vpuq du ds. (5.183)

Note that w is continuously differentiable and observe that w1 : r0, T s Ñ R is abso-lutely continuous. Moreover, note that (ii) implies that for all t P r0, T s it holds that

wptq “

ż T

0


Combining this with (5.182) proves that for Borelr0,T s-almost all t P r0, T s it holdsthat

µ ¨ vptq “ wptq. (5.185)


Next note that (5.185) implies that if µ “ 0, then it holds for all t P r0, T s that0 “ wptq “ w1ptq “ w2ptq and this shows that for Borelr0,T s-almost all t P r0, T s itholds that

0 “ w2ptq “ ´vptq. (5.186)

Equation (5.186) contradicts to the assumption thatşT

0|vpsq|2 ds “ 1 ą 0 and this

proves that µ ‰ 0. Next let v : p0, T q Ñ R be a continuously differentiable functiondefined by vptq :“ 1

µwptq for all t P r0, T s. Equation (5.185) then shows that for

Borelr0,T s-almost all t P p0, T q it holds that

vptq “ vptq (5.187)

and (5.183) and (5.184) hence imply that for all t P r0, T s it holds that

µ ¨ vptq “ wptq “

ż t

0

ż T

s

vpuq du ds “

ż T

0


This proves that w and v are twice continuously differentiable with the property thatfor all t P r0, T s it holds that

v2ptq “1

µw2ptq “

´1

µvptq. (5.189)

Exercise 5.5.18 hence proves that there exist A,B P C such that for all t P r0, T s itholds that vptq “ A exp

`

ta

´1µ˘

` B exp`

´ ta

´1µ˘

. This together with the fact

that vp0q “ wp0qµ“ 0 shows that for all t P r0, T s it holds that

vptq “ A”

exp´

ta

´1µ

¯

´ exp´

´ta

´1µ

¯ı

. (5.190)

The identity v1pT q “ w1pT qµ“ 0 and the assumption that

şT

0|vpsq|2 ds “ 1 ą 0 hence

prove that

exp´

2Ta

´1µ

¯

“ ´1. (5.191)

This implies that µ ą 0 and that there exists a k P N such that 2Ta

1µ “ 2πk ´ π.Hence, we obtain that

µ “ T 2

π2rk´12s2 . (5.192)

Putting this into (5.190) proves that for all t P r0, T s it holds that

vptq “ 2iA sin`“

k ´ 12

‰

tπT

˘

. (5.193)


This and the assumption thatşT

0|vpsq|2 ds “ 1 implies that

1 “

ż T

0

|vpsq|2 ds “ 4 |A|2ż T

0

ˇ

ˇsin`“

k ´ 12

‰

sπT

˘ˇ

ˇ

2ds

“ 2T |A|2ż 1

0

ˇ

ˇ

?2 sin

`

rk ´ 12s πs

˘ˇ

ˇ

2ds “ 2T |A|2 .

(5.194)

This and (5.193) prove that there exists a z P t´1, 1u such that for all t P r0, T s itholds that

vptq “ z?

2?T

sin`“

k ´ 12

‰

tπT

˘

“ z ekptq. (5.195)

Furthermore, observe that for all k P N, t P r0, T s it holds that

d2

dt2

„ż T

0

mints, tu ekpsq ds

“d2

dt2

„ż t

0

ż T

s

ekpuq du ds

“d

dt

„ż T

t

ekpsq ds

“ ékptq

“ T 2

π2rk´12s2 ¨ e

2kptq “

d2

dt2

”

T 2

π2rk´12s2 ¨ ekptq

ı

.

(5.196)

This together with the fact that for all k P N it holds that

T 2

π2rk´12s2 ¨ ekp0q “ 0 “

ż T

0

minps, 0q ekpsq ds and

T 2

π2rk´12s2 ¨ e

1kpT q “

Tπrk´12s

¨?

2?T¨ cos

`“

k ´ 12

‰

TπT

˘

“ T?

2πrk´12s

?T¨ cos

`“

k ´ 12

‰

π˘

“ 0 “

„

d

dt

ż t

0

ż T

s

ekpuq du ds

t“T

“

„

d

dt

ż T

0

minps, tq ekpsq ds

t“T

(5.197)

proves that for all k P N it holds that

T 2

π2rk´12s2 ¨ ek “ CovOppW q ek. (5.198)

Combining this with (5.192) proves (iii). Moreover, (5.198), (5.195) and Theo-rem 5.4.11 imply (iv) and (v). Finally, (iv), (v) and Corollary 5.5.8 prove (vi)–(vii).The proof of Theorem 5.5.19 is thus completed.

1 function [ Preimage , BM] = KLE Brownian Motion (T,N, Grid )2 Preimage = ( 0 :T/Grid :T) ;3 BM = Preimage ∗0 ;


4 for n=1:N5 s q r t e i g e n v n = T/(n ´ 1/2)/ pi ;6 e i g e n f n = sqrt (2/T)∗ sin ( Preimage/ s q r t e i g e n v n ) ;7 BM = BM + s q r t e i g e n v n ∗ e i g e n f n ∗ randn ;8 end9 end

Matlab code 5.1: A Matlab function for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 10 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;

10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f

Matlab code 5.2: A Matlab code for the approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.


10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;


12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f

Matlab code 5.3: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.


10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f

Matlab code 5.4: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

Figure 5.1: Results of calls of the Matlab codes 5.2–5.4.


Chapter 6

Stochastic processes with values ininfinite dimensional spaces

6.1 Hilbert space valued stochastic processes

6.1.1 Filtrations

Definition 6.1.1 (Filtration*). Let pΩ,Fq be a measurable space. Then we say thatF is a filtration on pΩ,Fq if and only if it holds

(i) that F is a mapping,

(ii) that DpFq Ď r´8,8s,

(iii) that codomainpFq “ PpΩq, and

(iv) that for all t1, t2 P DpFq with t1 ď t2 it holds that σΩpFt1q “ Ft1 Ď Ft2 Ď F.

Definition 6.1.2 (Filtrations associated to a filtration*). Let pΩ,Fq be a measurablespace and let F be a filtration on pΩ,Fq. Then we denote by F´ P MpDpFq,PpΩqqand F` PMpDpFq,PpΩqq the filtrations on pΩ,Fq which satisfy for all t P DpFq that

F´t “

#

σΩ

`

YsPTXr´8,tq Fs˘

: t ą infpTq

Ft : t “ infpTq(6.1)

and

F`t “

#

XsPTXpt,8s Fs : t ă suppTq

Ft : t “ suppTq. (6.2)

187

188 CHAPTER 6. STOCHASTIC PROCESSES

Lemma 6.1.3 (Properties of the filtrations associated to a filtration*). Let pΩ,Fqbe a measurable space and let F be a filtration on pΩ,Fq. Then

(i) it holds for all t P DpFq that F´t Ď Ft Ď F`t ,

(ii) it holds for all s, t P DpFq with s ă t that Fs Ď F´t Ď Ft, and

(iii) it holds for all s, t P DpFq with s ą t that Ft Ď F`t Ď Fs.

Proof of Lemma 6.1.3*. Items (i)–(iii) are an immediate consequence of Definition 6.1.1,(6.1), and (6.2). The proof of Lemma 6.1.3 is thus completed.

Lemma 6.1.4 (Further properties of the filtrations associated to a filtration*). LetpΩ,Fq be a measurable space, let a P r´8,8s, b P ra,8s, and let pFtqtPra,bs be afiltration on pΩ,Fq. Then it holds for all t P ra, bs that

`

pF´s qsPra,bs˘´

t“ F´t and

`

pF`s qsPra,bs˘`

t“ F`t . (6.3)

Proof of Lemma 6.1.4*. Throughout this proof assume w.l.o.g. that a ă b. Next notethat item (ii) of Lemma 6.1.3 ensures that for all t P pa, bs, r P ra, tq “ ra, bsXr´8, tqit holds that

Fr Ď pYsPra,bsXr´8,tqF´s q. (6.4)

This implies for all t P pa, bs that

pYsPra,bsXr´8,tqFsq Ď pYsPra,bsXr´8,tqF´s q. (6.5)

Hence, we obtain for all t P pa, bs that

F´t “ σΩ

`

YsPra,bsXr´8,tqFs˘

“ σΩ

`

YsPra,bsXr´8,tqF´s˘

“`

pF´s qsPra,bs˘´

t. (6.6)

Next observe that Item (iii) in Lemma 6.1.3 shows that for all t P ra, bq, r P pt, bs “ra, bs X pt,8s it holds that

pXsPra,bsXpt,8sF`s q Ď Fr. (6.7)

This implies for all t P ra, bq that

`

pF`s qsPra,bs˘`

t“ pXsPra,bsXpt,8sF`s q “ pXsPra,bsXpt,8sFsq “ F`t . (6.8)

Combining (6.6) and (6.8) completes the proof of Lemma 6.1.4.

6.1. HILBERT SPACE VALUED STOCHASTIC PROCESSES 189

Definition 6.1.5 (Left-continuity of a filtration*). Let pΩ,Fq be a measurable space.Then we say that F is a left-continuous filtration on pΩ,Fq if and only if F is afiltration on pΩ,Fq which satisfies for all t P DpFq that Ft “ F´t .

Definition 6.1.6 (Right-continuity of a filtration*). Let pΩ,Fq be a measurablespace. Then we say that F is a right-continuous filtration on pΩ,Fq if and only if Fis a filtration on pΩ,Fq which satisfies for all t P DpFq that Ft “ F`t .

Let pΩ,Fq, let T Ď r´8,8s be a set, and let pFtqtPT be a filtration on pΩ,FqThen, in general, it does not hold that for all t P T with t ą infpTq it holds thatF´t “ YsPTXp´8,tqFs because, in general, it does not hold that for all t P T witht ą infpTq it holds that YsPTXp´8,tqFs is a sigma-algebra. This is illustrated in thenext example.

Example 6.1.7. Let pΩ,Fq be the measurable space given by Ω “ N0 “ t0, 1, 2, . . . uand F “ PpΩq, let T Ď R be the set given by T “ r0, 1s, and let Ft Ď PpΩq, t P T,be the sets which satisfy for all n P N0, t P r1´ 12n, 1´ 12pn`1qq that

Ft “ σΩ

`

t0u, t1u, t2u, . . . , tnu(˘

(6.9)

and F1 “ PpΩq. Then

(i) observe

• that for all t P r0, 12q it holds that Ft “ σΩ

`

t0u(˘

,

• that for all t P r12, 3

4q it holds that Ft “ σΩ

`

t0u, t1u(˘

,

• that for all t P r34, 7

8q it holds that Ft “ σΩ

`

t0u, t1u, t2u(˘

,

• . . . ,

(ii) observe that pFtqtPT is a right-continuous filtration on pΩ,Fq,

(iii) observe that pFtqtPT is not a left-continuous filtration on pΩ,Fq,

(iv) observe that for all n P Ω it holds that

tnu P YsPTXp´8,1qFs “ YsPr0,1qFs, (6.10)

(v) observe that

YnPt0,2,4,6,... utnu “ t0, 2, 4, 6, . . . u R YsPTXp´8,1qFs “ YsPr0,1qFs, (6.11)

and

(vi) observe that YsPTXp´8,1qFs “ YsPr0,1qFs is not a sigma-algebra.


Class exercise 6.1.8 (*). Let pΩ,Fq be a measurable space, let T Ď r´8,8s be aset, and let pFtqtPT be a filtration on pΩ,Fq.

(i) Is pF´t qtPT a left-continuous filtration on pΩ,Fq?

(ii) Is pF`t qtPT a right-continuous filtration on pΩ,Fq?

Definition 6.1.9 (*). A quadrupel pΩ,F ,P,Fq is called a filtered probability space ifand only if it holds

(i) that pΩ,F ,Pq is a probability space and

(ii) that F is a filtration on pΩ,Fq.

Next we present the notions of a normal filtration (cf., e.g., Definition 2.1.11 in[18]) and of a stochastic basis (cf. Appendix E in [18]).

Definition 6.1.10 (Normal filtration*). Let pΩ,F ,Pq be a probability space. Thenwe say that F is a normal filtration on pΩ,F ,Pq if and only if it holds

(i) that F is a right-continuous filtration on pΩ,Fq and

(ii) that tA P F : PpAq “ 0u Ď pXtPDpFqFtq.

Definition 6.1.11 (Stochastic basis*). A quadrupel pΩ,F ,P,Fq is called a stochasticbasis if and only if it holds

(i) that pΩ,F ,P,Fq is a filtered probability space and

(ii) that F is a normal filtration on pΩ,F ,Pq.

Let a P r´8,8s, b P ra,8s, let pΩ,F ,Pq be a probability space, and let pFtqtPra,bsbe a filtration on pΩ,Fq. Then sometimes the quadrupel pΩ,F ,P, pFtqtPra,bsq is calleda stochastic basis in the literature although pFtqtPra,bs is not necessarily normal.

Proposition 6.1.12 (Construction of a stochastic basis*). Let pΩ,Fq be a measur-able space, let a P r´8,8s, b P ra,8s, let pFtqtPra,bs be a filtration on pΩ,Fq, and letGt Ď PpΩq, t P ra, bs, be the mapping with the property that for all t P ra, bs it holdsthat

Gt “ σΩpFt Y tA P F : PpAq “ 0uq . (6.12)

Then


(i) it holds that pΩ,F ,P, pG`t qtPra,bsq is a stochastic basis and

(ii) it holds for all normal filtrations pHtqtPra,bs on pΩ,F ,Pq with @ t P ra, bs : Ft ĎHt that @ t P ra, bs : G`t Ď Ht.

Proof of Proposition 6.1.12*. First, observe that (6.12) ensures that for all t P ra, bsit holds that

tA P F : PpAq “ 0u Ď Gt. (6.13)

Item (i) in Lemma 6.1.3 hence ensures that for all t P ra, bs it holds that

tA P F : PpAq “ 0u Ď G`t . (6.14)

Combining this with Lemma 6.1.4 establishes that pG`t qtPra,bs is a normal filtra-tion on pΩ,F ,Pq. This proves (i). Next observe that (6.12) ensures that for allnormal filtrations pHtqtPra,bs on pΩ,F ,Pq with @ t P ra, bs : Ft Ď Ht it holds that@ t P ra, bs : Gt Ď Ht. This implies that for all normal filtrations pHtqtPra,bs on pΩ,F ,Pqwith @ t P ra, bs : Ft Ď Ht it holds that

@ t P ra, bs : G`t Ď H`t “ Ht. (6.15)

This establishes (ii). The proof of Proposition 6.1.12 is thus completed.

6.1.2 Standard Wiener processes

Definition 6.1.13 (*). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbertspace, let Q P L1pHq be nonnegative and symmetric, and let pΩ,F ,P, pFtqtPr0,T sq bea filtered probability space. Then we say that W is a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process if and only if it holds that W PMpr0, T sˆΩ, Hq is an pFtqtPr0,T s/BpHq-adapted stochastic process which satisfies

(i) that PpW0 “ 0q “ 1,

(ii) that there exists a set A P tB P F : PpBq “ 1u such that for all ω P A it holdsthat r0, T s Q t ÞÑ Wtpωq P H is continuous,

(iii) that for all t1, t2 P r0, T s with t1 ď t2 it holds that σΩpWt2 ´Wt1q and Ft1 areP-independent, and

(iv) that for all t1, t2 P r0, T s with t1 ď t2 it holds that pWt2 ´ Wt1qpPqBpHq “N0,Qpt2´t1q.


Definition 6.1.14 (*). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space,let Q P L1pHq be nonnegative and symmetric, and let pΩ,F ,Pq be a probability space.Then we say that W is a Q-standard P-Wiener process if and only if it holds (thatW P Mpr0, T s ˆ Ω, Hq and that W is a Q-standard pΩ,F ,P, pFWt qtPr0,T sq-Wienerprocess).

Theorem 6.1.15 (*). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, andlet Q P L1pHq be nonnegative and symmetric. Then there exist a probability spacepΩ,F ,Pq and a Q-standard P-Wiener process W : r0, T s ˆ Ω Ñ H.

Theorem 6.1.15 can, e.g., be proved by using a Karhunen-Loeve expansion similaras in the proof of Theorem 5.5.11.

Proposition 6.1.16 (Normalization*). Let T P r0,8q, let pΩ,F ,P, pFtqtPr0,T sq bea filtered probability space, let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, letQ P L1pHq be nonnegative and symmetric, let W : r0, T s ˆ Ω Ñ H be a Q-standardpΩ,F ,P, pFtqtPr0,T sq-Wiener process, and let Gt Ď PpΩq, t P r0, T s, be the mappingwith the property that for all t P r0, T s it holds that

Gt “ σΩpFt Y tA P F : PpAq “ 0uq . (6.16)

Then

(i) it holds that pΩ,F ,P, pG`t qtPr0,T sq is a stochastic basis,

(ii) it holds for all normal filtrations pHtqtPr0,T s on pΩ,F ,Pq with @ t P r0, T s : Ft ĎHt that @ t P r0, T s : G`t Ď Ht, and

(iii) it holds that W is a Q-standard pΩ,F ,P, pG`t qtPr0,T sq-Wiener process.

Proof of Proposition 6.1.16*. First, observe that Item (i) and Item (ii) follow imme-diately from Items (i)–(ii) in Proposition 6.1.12. It thus remains to prove Item (iii).For this note that Definition 6.1.13 ensures that for all t1, t2 P r0, T s with t1 ď t2 itholds that σΩpWt2 ´Wt1q and Ft1 are P-independent. Hence, we obtain that for allt1 P r0, T s, t2 P rt1, T s, A P Ft1 , W P σΩpWt2 ´Wt1q it holds that

PpAXWq “ PpAq ¨ PpWq. (6.17)

This ensures that for all t1 P r0, T s, t2 P rt1, T s, A P Ft1 , B P tS P F : PpSq P t0, 1uu,W P σΩpWt2 ´Wt1q it holds that

P`

pAXBq XW˘

“ P`

pAXWq XB˘

“

#

PpAXWq : PpBq “ 1

0 : PpBq “ 0

“ PpAXBq ¨ PpWq.(6.18)


Next observe that for all t P r0, T s it holds that

Gt “ σΩ

´

Ft Y σΩ

`

A P F : PpAq “ 0(˘

¯

“ σΩ

´

Ft Y

A P F : PpAq P t0, 1u(

¯

“ σΩ

´!

pAXBq P PpΩq : A P Ft and B P

S P F : PpSq P t0, 1u(

)¯

.

(6.19)

Combining this and the fact that for every t P r0, T s it holds that the set tpA XBq P PpΩq : A P Ft and B P tS P F : PpSq P t0, 1uuu is X-stable with (6.18) andLemma 5.1.10 establishes that for all t1 P r0, T s, t2 P rt1, T s it holds that Gt1 andσΩpWt2 ´Wt1q are P-independent. This implies that for all t1 P r0, T s, t2 P rt1, T s,A P Gt1 , ϕ P CpH,Rq, ψ P CpR,Rq with sup

`

t|ϕpxq| : x P HuYt|ψpxq| : x P Ru˘

ă 8

it holds that

ErϕpWt2 ´Wt1qψp1Aqs “ ErϕpWt2 ´Wt1qs Ërψp1Aqs . (6.20)

Hence, we obtain that for all t1 P r0, T q, t2 P pt1, T s, n P N X p1t2´t1,8q, A P G`t1 ,ϕ P CpH,Rq, ψ P CpR,Rq with sup

`

t|ϕpxq| : x P HuYt|ψpxq| : x P Ru˘

ă 8 it holdsthat

E“

ϕpWt2 ´Wt1`1nqψp1Aq‰

“ E“

ϕpWt2 ´Wt1`1nq‰

Ërψp1Aqs . (6.21)

Lebesgue’s theorem of dominated convergence therefore shows that for all t1 P r0, T q,t2 P pt1, T s, A P G`t1 , ϕ P CpH,Rq, ψ P CpR,Rq with sup

`

t|ϕpxq| : x P Hu Yt|ψpxq| : x P Ru

˘

ă 8 it holds that

ErϕpWt2 ´Wt1qψp1Aqs “ lim supnÑ8

E“

ϕpWt2 ´Wt1`1nqψp1Aq‰

“ lim supnÑ8

`

E“

ϕpWt2 ´Wt1`1nq‰

Ërψp1Aqs˘

“ ErϕpWt2 ´Wt1qs Ërψp1Aqs .

(6.22)

Corollary 5.3.26 hence shows that for all t1 P r0, T q, t2 P pt1, T s, A P G`t1 it holds thatWt2 ´Wt1 and Ω Q ω ÞÑ 1Apωq P R are P-independent. This ensures that for allt1 P r0, T q, t2 P pt1, T s, A P G`t1 , W P BpHq it holds that

P`

AX tWt2 ´Wt1 PWu˘

“ PpAq ¨ PpWt2 ´Wt1 PWq. (6.23)

This shows that for all t1 P r0, T q, t2 P pt1, T s it holds that G`t1 and σΩpWt2 ´Wt1q

are P-independent. This establishes (iii). The proof of Proposition 6.1.16 is thuscompleted.


6.1.3 Pseudo inverse

Lemma 6.1.17 (*). Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbertspaces, and let A P LpH1, H2q. Then

(i) it holds that the mapping A|KernpAqK : KernpAqK Ñ H2 is injective and

(ii) it holds that impAq “ im`

A|KernpAqK˘

.

Proof of Lemma 6.1.17*. First of all, recall that

KernpAqK “

v P H1 :“

@u P KernpAq : 〈v, u〉H1“ 0

‰(

(6.24)

is a K-vector subspace of H1. The mapping A|KernpAqK : KernpAqK Ñ H2 is thus alinear mapping from KernpAqK to H2. It thus holds that A|KernpAqK is injective if andonly if

Kern`

A|KernpAqK˘

“ t0u. (6.25)

Next note that

Kern`

A|KernpAqK˘

“

v P KernpAqK : A|KernpAqKpvq “ 0(

“

v P KernpAqK : Av “ 0(

“

v P KernpAqK : v P KernpAq(

“ KernpAqK XKernpAq “ t0u.

(6.26)

Combining this with (6.25) proves that A|KernpAqK is injective. Moreover, observethat

impAq “ ApH1q “ tAv P H2 : v P H1u

“

A“

PKernpAq,H1 rvs ` PKernpAqK,H1rvs

‰

P H2 : v P H1

(

“

APKernpAqK,H1rvs P H2 : v P H1

(

“

APKernpAqK,H1rvs P H2 : v P KernpAqK

(

“

Av P H2 : v P KernpAqK(

“ im`

A|KernpAqK˘

.

(6.27)


Lemma 6.1.17 allows us to introduce the following concept.

Definition 6.1.18 (*). Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces, and let A P LpH1, H2q. Then we denote by A´1 : impAq Ñ H1 thelinear operator with the property that for all v P impAq it holds that

A´1pvq “ A|´1KernpAqK

pvq (6.28)

and we call A´1 the pseudo inverse of A.


Class exercise 6.1.19 (*). Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, beK-Hilbert spaces, and let A P LpH1, H2q. What is impA´1q?

The next exercise illustrate the notion of the pseudo inverse in a specific exampleand, in particular, helps us to get more familiar with the concept of the pseudoinverse.

Exercise 6.1.20 (*). Let A : L2pBorelp0,1q; |¨|Rq Ñ R be the linear mapping with theproperty that for all v P L2pBorelp0,1q; |¨|Rq it holds that

Av “

ż 1

0

vpxq dx. (6.29)

Specify DpA´1q, impA´1q, codomainpA´1q, and A´1v, v P DpA´1q, explicity. Showthat your specifications are indeed correct.

In the following proposition we present an important property of the pseudoinverse of a bounded linear operator.

Proposition 6.1.21 (Minimality property of the pseudo inverse). Let K P tR,Cu,let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces, let A P LpH1, H2q, and letv P impAq “ DpA´1q. Then A´1vH1

“ infuPA´1ptvuq uH1and

A´1v(

“

#

w P H1 :

«

Aw “ v and wH1“ inf

uPH1,Au“v

uH1

ff+

“

"

w P A´1ptvuq : wH1

“ infuPA´1ptvuq

uH1

*

“ A´1ptvuq X

“

KernpAqK‰

.

(6.30)

Proof of Proposition 6.1.21. First of all, note that Lemma 6.1.17 and the definitionof the pseudo inverse prove that

A´1v(

“

!

A|´1KernpAqK

pvq)

“

w P KernpAqK : Aw “ v(

““

KernpAqK‰

X A´1ptvuq.

(6.31)Next recall that KernpAq Ď H1 is a closed subspace of H1. Definition 3.4.12 thusshows that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that

A|KernpAqK“

PKernpAqK rws‰

“ A“

PKernpAqK rws‰

“ A“

w ´ PKernpAq rws‰

“ Aw “ v.(6.32)


The fact that A|KernpAqK is injective (see Lemma 6.1.17) hence proves that for allw P A´1ptvuq “ tu P H1 : Au “ vu it holds that

PKernpAqK rws “ A´1v. (6.33)

This implies that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that

wH1“›

›PKernpAq rws ` PKernpAqK rws›

›

H1

“

b

›

›PKernpAq rws›

›

2

H1`›

›PKernpAqK rws›

›

2

H1

“

b

›

›PKernpAq rws›

›

2

H1` A´1pvq2H1

ě›

›A´1pvq

›

›

2

H1.

(6.34)

This and the fact that A´1pvq P A´1ptvuq prove that

infuPA´1ptvuq

uH1“›

›A´1pvq

›

›

H1. (6.35)

This, (6.34), and (6.31) imply that"

w P A´1ptvuq : wH1

“ infuPA´1ptvuq

uH1

*

“

!

w P A´1ptvuq : wH1

“›

›A´1pvq

›

›

H1

)

“

"

w P A´1ptvuq :

b

›

›PKernpAq rws›

›

2

H1` A´1pvq2H1

“›

›A´1pvq

›

›

H1

*

“

!

w P A´1ptvuq :

›

›PKernpAq rws›

›

2

H1“ 0

)

“

w P A´1ptvuq : PKernpAq rws “ 0

(

“ A´1ptvuq X

“

KernpAqK‰

“

A´1pvq

(

.

(6.36)

This, (6.31), and (6.35) complete the proof of Proposition 6.1.21.

The pseudo inverse allows us to define a Hilbert space structure on the imageof a bounded linear operator on Hilbert spaces. This is the subject of the nextproposition.

Proposition 6.1.22 (Image Hilbert space*). LetK P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq,k P t1, 2u, be K-Hilbert spaces, and let A P LpH1, H2q. Then it holds that

`

impAq,⟨A´1p¨q, A´1p¨q

⟩H1,›

›A´1p¨q›

›

H1

˘

(6.37)

is an K-Hilbert space.

6.2. SPACE-TIME WHITE NOISE AND BROWNIAN SHEET 197

Proof of Proposition 6.1.22*. First of all, note that the fact that A´1 : impAq Ñ H1

is a linear operator implies that

`

impAq,⟨A´1p¨q, A´1p¨q

⟩H1,›

›A´1p¨q›

›

H1

˘

(6.38)

is aK-inner product space. It thus remains to prove that pimpAq, A´1p¨qH1q is com-

plete. To see this let pvnqnPN Ď impAq be a Cauchy sequence in pimpAq, A´1p¨qH1q.

Hence, we obtain that A´1pvnq P KernpAqK, n P N, is a Cauchy sequence inpKernpAqK, ¨H1

q. Completeness of pKernpAqK, ¨H1q hence proves that there ex-

ists a vector w P KernpAqK such that

lim supnÑ8

›

›w ´ A´1pvnq›

›

H1“ 0. (6.39)

This proves that Aw P impAq satisfies limnÑ8 A´1pAw ´ vnqH1

“ 0. The proof ofProposition 6.1.22 is thus completed.

6.2 Space-time white noise and Brownian sheet

6.2.1 Derivative of a Brownian sheet

Definition 6.2.1 (Law of white noise). Let d,m P N and let D Ď Rd be an openset. Then we say that µ is the law of white noise on L2pD;Rmq if and only ifµ P M

`

BpDpD,Rmq1q, r0,8s˘

is a Gaussian measure which satisfies for all v, w P

DpD,Rmq that

ż

DpD,Rmq1φpvq ` φpvqφpwqµpdφq “ 〈v, w〉L2pBorelD;Rmq . (6.40)

Definition 6.2.2 (White noise). Let pΩ,F ,Pq be a probability space, let d,m P

N, and let D Ď Rd be an open set. Then we say that X is a P-white noise onL2pD;Rmq if and only if (it holds that X PMpF ,BpDpD,Rmq1qq and it holds thatXpPqBpDpD,Rmq1q is the law of white noise on L2pD,Rmqq.

Remark 6.2.3 (Space-time white noise). Let pΩ,F ,Pq be a probability space, letd P t2, 3, . . . u, m P N, and let D Ď Rd be an open set. Then in the literature X issometimes referred to as a space-time white noise on L2pD;Rmq with respect to P ifand only if X is a P-space-time white noise on L2pD;Rmq.


Theorem 6.2.4 (“Coordinate free” property of white noise). Let pΩ,F ,Pq be aprobability space, let d,m P N, let D Ď Rd be an open set, and let X : Ω Ñ DpD,Rmq1

be a F/BpDpD,Rmq1q-measurable mapping. Then X is a P-white noise on L2pD;Rmq

if and only if for all orthonormal basis H Ď L2pD;Rmq of L2pD;Rmq there existsa family ξh PMpF ,BpRqq, h P H, of i.i.d. standard normal random variables suchthat

X “ÿ

hPH

ξh h. (6.41)

Theorem 6.2.5 (Distributional derivative of Brownian motion). Let m P N, letpΩ,F ,Pq be a probability space, let W : r0, T s ˆ Ω Ñ Rm be a standard Brownianmotion with continuous sample paths, and let X : Ω Ñ Dpp0, T q;Rmq1 be the functionwhich satisfies for all ω P Ω, ϕ P Dpp0, T q,Rmq that

`

Xpωq˘

pϕq “ ´

ż T

0

〈ϕ1ptq,Wtpωq〉Rm dt “`

BWp¨qpωq˘

pϕq. (6.42)

Then X is a P-white noise on L2pp0, T q;Rmq.

6.3 Stochastic integration with respect to infinite

dimensional Wiener processes

The following presentations are similiar to the presentations in Section 2.3 in Prevot& Rockner [18] and in Chapter 3 in the lecture notes of the course Numerical Analysisof Stochastic Ordinary Differential Equations.

6.3.1 Lenglart’s inequality

Definition 6.3.1 (Random time). Let T Ď pRYt´8,8uq be a set, let pΩ,F ,Pq bea probability space, and let τ : Ω Ñ T be an F/BpTq-measurable mapping. Then τis called a random time.

Observe, in the setting of Definition 6.3.1, that for every t P T it holds thattτ ď tu P F .

Definition 6.3.2 (Stopping time). Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pqbe a probability space with a filtration pFtqtPT, and let τ : Ω Ñ T be a mappingwith the property that for all t P T it holds that tτ ď tu P Ft. Then τ is called anpFtqtPT-stopping time.

6.3. STOCHASTIC INTEGRATION 199

A stopping time on a filtered probability space also induces a sigma-algebra. Thisis the subject of the next definition.

Definition 6.3.3. Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pq be a probabilityspace with a filtration pFtqtPT, and let τ : Ω Ñ T be an pFtqtPT-stopping time. Thenwe denote by Fτ the set given by

Fτ “ tA P pYtPTFtq : p@ t P T : AX tτ ď tu P Ftqu (6.43)

and we call Fτ the sigma-algebra at the stopping time τ .

Exercise 6.3.4. Let T Ď pRYt´8,8uq be a set, let pΩ,F ,Pq be a probability spacewith a filtration pFtqtPT, and let τ, ρ : Ω Ñ T be pFtqtPT-stopping times. Prove thenthat mintτ, ρu is an pFtqtPT-stopping time.

In (6.46) in the following result, Proposition 6.3.5, we prove a powerful inequalitywhich is known as Lenglart inequality in the literature. Proposition 6.3.5 and itsproof are extensions of Problem 1.4.15, Remark 1.4.17 and Solution 4.15 in Section1.6 in [12].

Proposition 6.3.5 (Lenglart inequality). Let pΩ,F ,Pq be a probability space witha filtration pFtqtPr0,8q, let X, Y : r0,8qˆΩ Ñ r0,8q be pFtqtPr0,8q/Bpr0,8qq-adaptedstochastic processes with continuous sample paths such that for all bounded pFtqtPr0,8q-stopping times τ : Ω Ñ r0,8q it holds that E

“

Xτ

‰

ď E“

suptPr0,τ s Yt‰

. Then for allε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ r0,8q it holds that

P`

suptPr0,τ sXt ě ε˘

ď 1εE“

suptPr0,τ s Yt‰

, (6.44)

P`

suptPr0,τ sXt ě ε, suptPr0,τ s Yt ă δ˘

ď 1εE“

min

δ, suptPr0,τ s Yt(‰

, (6.45)

P`


ď 1εE“

min


` P`

suptPr0,τ s Yt ě δ˘

, (6.46)

E“

min

ε, suptPr0,τ sXt

(‰

ď

”

2?ε` ε?

δ

ı

ˇ

ˇE“

min

δ, suptPr0,τ s Yt(‰ˇ

ˇ

12, (6.47)

E“

min

1, suptPr0,τ sXt

(‰

ď 3ˇ

ˇE“

min

1, suptPr0,τ s Yt(‰ˇ

ˇ

12. (6.48)

Proof of Proposition 6.3.5. Throughout this proof let ρXε : Ω Ñ r0,8s, ε P r0,8q,and ρYε : Ω Ñ r0,8s, ε P r0,8q, be the mappings with the property that for allε P r0,8q it holds that

ρXε “ inf`

t P r0,8q : Xt ě ε(

Y t8u˘

, (6.49)


ρYε “ inf`

t P r0,8q : supsPr0,ts Ys ě ε(

Y t8u˘

. (6.50)

Then observe that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stopping times τ : Ω Ñr0,8q it holds that

εP`

suptPr0,mintτ,nusXt ě ε˘

“ εP´

D t P r0,mintτ, nus : Xt ě ε¯

“ εP´!

D t P r0,mintτ, nus : Xt ě ε)

X

!

ρXε ď mintτ, nu)¯

“ εP´!

D t P r0,mintτ, nus : Xt ě ε)

X

!

ρXε ď mintτ, nu)

X

!

Xmintτ,n,ρXε uě ε

)¯

ď εP´

Xmintτ,n,ρXε uě ε

¯

“ E

”

ε1tXmintτ,n,ρXε u

ěεu

ı

ď E

”

Xmintτ,n,ρXε u1tX

mintτ,n,ρXε uěεu

ı

ď E“

Xmintτ,n,ρXε u

‰

.

(6.51)

Combining this with the fact for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that mintτ, n, ρXε u is a bounded pFtqtPr0,8q-stopping time(see Exercise 6.3.4) ensures that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that

εP`


ď E“

Xmintτ,n,ρXε u

‰

ď E“

suptPr0,mintτ,n,ρXε usYt‰

ď E“

suptPr0,τ s Yt‰

.(6.52)

Hence, we obtain that for all ε P r0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ

r0,8q it holds that

εP`


“ εP`

YnPN

suptPr0,mintτ,nusXt ě ε(˘

“ ε limnÑ8

P`


ď E“

suptPr0,τ s Yt‰

.(6.53)

This proves (6.44). In the next step we observe that (6.44) ensures that for allε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`


“ P`

suptPr0,τ sXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ˘

“ P´

suptPr0,mintτ,ρYδ usXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ

¯

ď P´

suptPr0,mintτ,ρYδ usXt ě ε

¯

ď 1εE

”

suptPr0,mintτ,ρYδ usYt

ı

“ 1εE

”

min

δ, suptPr0,mintτ,ρYδ usYt(

ı

ď 1εE“

min


.

(6.54)


This proves (6.45). Furthermore, we observe that (6.45) shows that for all ε, δ Pp0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`


ď P`


` P`


ď 1εE“

min


` P`


.

(6.55)

This proves (6.46). Next we note that (6.46) and the Markov inequality show thatfor all r, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`

suptPr0,τ sXt ě r˘

ď 1rE“

min


` P`

min

δ, suptPr0,τ s Yt(

ě δ˘

ď“

1r` 1

δ

‰

E“

min


.

(6.56)

This implies that for all ε, δ, r P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ

p0,8q it holds that

E“

min

ε, suptPr0,τ sXt

(‰

“ E

”

min

ε, suptPr0,τ sXt

(

1tsuptPr0,τsXtăru

ı

È

”

min

ε, suptPr0,τ sXt

(

1tsuptPr0,τsXtěru

ı

ď mintε, ru ` εP`

suptPr0,τ sXt ě r˘

ď mintε, ru ` ε“

1r` 1

δ

‰

E“

min


ď r ` ε“

1r` 1

δ

‰

E“

min


.

(6.57)

Hence, we obtain that for all ε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñp0,8q it holds that

E“

min

ε, suptPr0,τ sXt

(‰

ď infrPp0,8q

`

r ` εrE“

min


` εδE“

min

δ, suptPr0,τ s Yt(‰˘

ďˇ

ˇεE“

min


ˇ

12

`?εˇ

ˇE“

min


ˇ

12` ε

δE“

min


.

(6.58)

This proves (6.47). Moreover, we note that (6.48) is an immediate consequence of(6.47). The proof of Proposition 6.3.5 is thus completed.


Exercise 6.3.6 (Characterization of convergence in probability). Let pΩ,F ,Pq bea probability space, let pE, dEq be a metric space, and let Xn : Ω Ñ E, n P N0, beF/pE, dEq-strongly measurable mappings. Then the following three statements areequivalent:

(i) For all c P p0,8q it holds that

limnÑ8

E“

mintc, dEpX0, Xnqu‰

“ 0. (6.59)

(ii) There exists a c P p0,8q such that

limnÑ8

E“

mintc, dEpX0, Xnqu‰

“ 0. (6.60)

(iii) For all ε P p0,8q it holds that

limnÑ8

PpdEpX0, Xnq ą εq “ 0. (6.61)

6.3.2 Modifications and indistinguishability

This material is an extended version from Barth et al. 2014. Next we address twonotions that somehow describe when two stochastic processes are “equal up to setsof measure zero”.

Definition 6.3.7 (Modifications). Let pΩ,F ,Pq be a probability space, let pS,Sq bea measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S be stochasticprocesses such that for every t P T it holds that there exists an event A P F withPpAq “ 1 and

A Ď tXt “ Ytu. (6.62)

Then X and Y are called modifications of each other (i.e., X is called a modificationof Y and Y is called a modification of X).

Exercise 6.3.8. Prove or disprove the following statement: For all measurable spacespΩ,Fq it holds that tpω, ωq P Ω2 : ω P Ωu P F b F .

Exercise 6.3.9. Specify explicitly measurable spaces pΩ,Fq and pS,Sq and F/S-measurable mappings X, Y : Ω Ñ S such that tX “ Y u “ tω P Ω: Xpωq “ Y pωqu RF . Prove that your result is correct.


Definition 6.3.10 (Indistinguishablility). Let pΩ,F ,Pq be a probability space, letpS,Sq be a measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S bestochastic processes with the property that there exists an event A P F with PpAq “ 1and

A Ď pXtPTtXt “ Ytuq . (6.63)

Then X and Y are called indistinguishable from each other (i.e., X is called indis-tinguishable from Y and Y is called indistinguishable from X).

Let us illustrate Definitions 6.3.7 and 6.3.10 through a simple example (see, e.g.,Kuhn [15]).

Example 6.3.11. Let pΩ,F ,Pq be a probability space, let pFtqtPr0,1s be the filtrationon pΩ,Fq with the property that for all t P r0, 1s it holds that Ft “ F , let τ : Ω Ñ r0, 1sbe an F/Bpr0, 1sq-measurable mapping such that τpPq “ Borelr0,1s, let X, Y : r0, 1s ˆΩ Ñ R be the functions with the property that for all ω P Ω, t P r0, 1s it holds that

Xtpωq “ 0 and Ytpωq “

#

1 : t “ τpωq

0 : t ‰ τpωq. (6.64)

Then

(i) it holds that X, Y are pFtqtPr0,T s-predictable stochastic processes (indeed, letY n : r0, T s ˆ Ω Ñ R, n P N, be the mappings with the property that for alln P N, t P r0, T s it holds that Y n

t pωq “ 1pτpωq´1n,τpωqsptq, observe that forall n P N it holds that Y n is pFtqtPr0,T s-predictable and note that @ pt, ωq Pr0, T s ˆ Ω: limnÑ8 Y

nt pωq “ Ytpωq),

(ii) it holds that τ is an pFtqtPr0,T s-stopping time,

(iii) it holds for all ω P Ω that Xτpωqpωq “ 0 ‰ 1 “ Yτpωqpωq,

(iv) it holds that!

ω P Ω:`

@ t P r0, T s : Xtpωq “ Ytpωq˘

)

“

!

ω P Ω:`

@ t P r0, T s : Ytpωq “ 0˘

)

“ H,(6.65)

(v) it holds for all t P r0, T s that

PpXt “ Ytq “ PpYt “ 0q “ Ppτ ‰ tq “ 1, (6.66)

(vi) and it holds that X and Y are modification of each other but X and Y are notindistinguishable from of each other.


6.3.3 Predictability

Definition 6.3.12 (Predictable sigma-algebra*). Let T P r0,8q and let pΩ,Fq be ameasurable space with a filtration pFtqtPr0,T s. Then we denote by Pred

`

pFtqtPr0,T s˘

thesigma-algebra given by

Pred`

pFtqtPr0,T s˘

“

σr0,T sˆΩ

´

tps, ts ˆ A : A P Fs and s, t P r0, T s with s ă tu Y tt0u ˆ A : A P F0u

¯

(6.67)

and we call Pred`

pFtqtPr0,T s˘

the predictable sigma-algebra of pFtqtPr0,T s.

Note, in the setting of Definition 6.3.12, that the definition of the sigma-algebraPred

`

pFtqtPr0,T s˘

depends on the filtration pFtqtPr0,T s.

Definition 6.3.13 (Predictability*). Let T P r0,8q, let pS,Sq be a measurable space,and let pΩ,Fq be a measurable space with a filtration pFtqtPr0,T s. Then we say that Xis an pFtqtPr0,T s/S-predictable process (we say that X is an pFtqtPr0,T s/S-predictablestochastic process, we say X is a predictable process, we say that X is a predictablestochastic process) if and only if X P Mpr0, T s ˆ Ω, Sq is an Pred

`

pFtqtPr0,T s˘

/S-measurable mapping.

Observe that for every T P p0,8q and every measurable space pΩ,Fq with afiltration pFtqtPr0,T s it holds that

PredppFtqtPr0,T sq Ď σr0,T sˆΩ

`

B ˆ A : B P Bpr0, T sq and A P FT˘

“ Bpr0, T sq b FT .(6.68)

This is fact is used in the next definition.

Definition 6.3.14 (Product measure on the predictable sigma-algebra*). Let T Pp0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s. Then wedenote by

PP,pFtqtPr0,T s : PredppFtqtPr0,T sq Ñ r0,8s (6.69)

the measure given by

PP,pFtqtPr0,T s “ pBorelr0,T sbPq|PredppFtqtPr0,T sq. (6.70)

Let T P p0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s.Then we note that for all t1, t2 P r0, T s, A P Ft1 with t1 ă t2 it holds that

PP,pFtqtPr0,T sppt1, t2s ˆ Aq “ pt2 ´ t1q ¨ PpAq. (6.71)


Class exercise 6.3.15 (Measurability of sample paths?*). Let T P p0,8q, let pS,Sqbe a measurable space with a filtration pFtqtPr0,T s, and let X : r0, T s ˆ Ω Ñ S be anpFtqtPr0,T s/S-predictable stochastic process. Is it then true that for every ω P Ω itholds that

r0, T s Q t ÞÑ Xtpωq P S (6.72)

is a Bpr0, T sq/S-measurable mapping?

6.3.4 Construction of the stochastic integral

In the next step we introduce the notion of an elementary process. For this we recallthe notion of a simple function; see Definition 2.3.1 above.

Definition 6.3.16 (Elementary process). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq andpU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let pΩ,Fq be a measurable space with a filtrationpFtqtPr0,T s, and let X : r0, T s ˆ Ω Ñ LpU,Hq be a mapping with the property thatthere exist n P N, 0 ď t1 ă . . . ă tn ď T and for every k P t1, . . . , n ´ 1u anFtk/BpLpU,Hqq-simple function Yk : Ω Ñ LpU,Hq such that for all t P r0, T s it holdsthat

Xt “

n´1ÿ

k“1

Yk ¨ 1ptk,tk`1sptq . (6.73)

Then X is called pFtqtPr0,T s-elementary (or just elementary).

Elementary processes in the sense of Definition 6.3.16 are predictable in the senseof Definition 6.3.13. Let pU, 〈¨, ¨〉U , Üq be an R-Hilbert space, let T P r0,8q,let pΩ,F ,Pq be a probability space, let Q P L1pUq be nonnegative and symmetric,and let W : r0, T s ˆ Ω Ñ U be a Q-standard P-Wiener process. In the stochasticintegration theory with respect to the possibly infinite dimensional Q-standard P-Wiener process W the Hilbert space

´

Q12pUq,

⟨Q´

12p¨q, Q´

12p¨q⟩U,›

›Q´12p¨q›

›

2

U

¯

(6.74)

plays an important role. Recall that Q12 is defined according to Theorem 3.5.15and Q´12 is the pseudo inverse of Q12 (see Definition 6.1.18 above). According toProposition 6.1.22 the triple (6.74) is indeed anR-Hilbert space. Lemma 6.3.20 belowillustrate the appearence of the Hilbert space in (6.74) in the stochastic integrationtheory. Before we present Lemma 6.3.20, we note the following exercise and itspreceding remark.


Exercise 6.3.17 (Embedding of LpU,Hq intoHSpQ12pUq, Hq*). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Prove that A|Q12pUq P HSpQ

12pUq, Hq and

›

›A|Q12pUq

›

›

HSpQ12pUq,Hq“›

›AQ12›

›

HSpU,Hqď ALpU,Hq

›

›Q12›

›

HSpUqă 8. (6.75)

Remark 6.3.18 (Embedding of LpU,Hq intoHSpQ12pUq, Hq*). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Then we often simply write A as an abbreviation forA|Q12pUq.

Lemma 6.3.19. Let pU, 〈¨, ¨〉U , Üq be an R-Hilbert space and let Q P L1pUq. ThenQ is a diagonal linear operator if and only if Q is symmetric.

We are now ready to present Lemma 6.3.20, which illustrates the appearence ofthe Hilbert space in (6.74) in the stochastic integration theory.

Lemma 6.3.20 (Baby version of Ito’s isometry*). Let T P r0,8q, s P r0, T s, t Prs, T s, let pΩ,F ,P, pFtqtPr0,T q be a stochastic basis, let pU, 〈¨, ¨〉U , Üq be a separableR-Hilbert space, let A : Ω Ñ LpU,Hq be an Fs/BpLpU,Hqq-simple function, let Q PL1pUq be nonnegative and symmetric, and let W : r0, T s ˆ Ω Ñ U be a Q-standardpΩ,F ,P, pFtqtPr0,T sq-Wiener process. Then

E“

A pWt ´Wsq2H

‰

“ E

”

A2HSpQ12pUq,Hq

ı

pt´ sq . (6.76)

Proof of Lemma 6.3.20*. Throughout this proof let B Ď U be an orthonormal basisof U and let λ : BÑ r0,8q be a globally bounded function such that for all u P U itholds that

Qu “ÿ

bPB

λb 〈b, u〉U b. (6.77)

Next note that the fact that B Ď U is an orthonormal basis of U and the continuity


of A imply that

E“

A pWt ´Wsq2H

‰

“ E

»

–

›

›

›

›

›

A

˜

ÿ

bPB

〈b,Wt ´Ws〉U b

¸›

›

›

›

›

2

H

fi

fl

“ E

»

–

›

›

›

›

›

ÿ

bPB

〈b,Wt ´Ws〉U Ab

›

›

›

›

›

2

H

fi

fl

“ E

«

ÿ

b1,b2PB

〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉H

ff

“ÿ

b1,b2PB

Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉Hs .

(6.78)

Independency and the definition of a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener processhence assure that

E“

A pWt ´Wsq2H

‰

“ÿ

b1,b2PB

Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U sEr〈Ab1, Ab2〉Hs

“ÿ

bPB

E“

|〈b,Wt ´Ws〉U |2‰

E“

Ab2H‰

“ pt´ sqÿ

bPB

〈b,Qb〉U E“

Ab2H‰

“ pt´ sqÿ

bPλ´1pp0,8qq

〈b,Qb〉Ulooomooon

“λb

E“

Ab2H‰

.

(6.79)

This proves that

E“

A pWt ´Wsq2H

‰

“ pt´ sqE

»

–

ÿ

bPλ´1pp0,8qq

›

›

›Aa

λb b›

›

›

2

H

fi

fl

“ pt´ sqE

»

–

ÿ

bPλ´1pp0,8qq

›

›AQ12b›

›

2

H

fi

fl .

(6.80)

Next we observe that the set

Q12pbq P im

`

Q12˘

: b P λ´1pp0,8qq

(

(6.81)

is an orthonormal basis of the R-Hilbert space´

Q12pUq,

⟨Q´

12p¨q, Q´

12p¨q⟩U,›

›Q´12p¨q›

›

2

U

¯

. (6.82)


Combining this with (6.80) completes the proof of Lemma 6.3.20.

Lemma 6.3.21 (Ito’s isometry in infinite dimension for elementary processes). Letn P t2, 3, . . . u, T P r0,8q, 0 ď t1 ă ¨ ¨ ¨ ă tn “ T , let pΩ,F ,P, pFtqtPr0,T q bea stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbertspaces, for every k P t1, 2, . . . , n ´ 1u let Yk : Ω Ñ LpU,Hq be an Ftk/BpLpU,Hqq-simple function, let Q P L1pUq be nonnegative and symmetric, and let W : r0, T s ˆΩ Ñ U be a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process. Then

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl “ E

«

n´1ÿ

k“1

Yk2HSpQ12pUq,Hq ptk`1 ´ tkq

ff

“

ż T

0

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk 1ptk,tk`1spsq

›

›

›

›

›

2

HSpQ12pUq,Hq

fi

fl ds.

(6.83)

Proof of Lemma 6.3.21. Note that

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl

“

n´1ÿ

k,l“1

E“⟨Yk

`

Wtk`1´Wtk

˘

, Yl`

Wtl`1´Wtl

˘⟩H

‰

“ 2ÿ

k,lPt1,...,n´1ukăl

E“⟨rYls

˚Yk`

Wtk`1´Wtk

˘

,Wtl`1´Wtl

⟩U

‰

`

n´1ÿ

k“1

E

”

›

›Yk`

Wtk`1´Wtk

˘›

›

2

H

ı

.

(6.84)

Independency and Lemma 6.3.20 hence imply that for all orthonormal bases B Ď U


of U it holds that

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl

“ 2ÿ


E

«

ÿ

bPB

⟨b, rYls

˚Yk`

Wtk`1´Wtk

˘⟩U

⟨b,Wtl`1

´Wtl

⟩U

ff

`

n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq

“ 2ÿ


ÿ

bPB

E“⟨b, rYls

˚Yk`

Wtk`1´Wtk

˘⟩U

‰

E“⟨b,Wtl`1

´Wtl

⟩U

‰

`

n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq “n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq .

(6.85)


The random variableřn´1k“1 Yk

`

Wtk`1´Wtk

˘

in (6.83) will be the stochastic in-

tegral of the elementary stochastic processřn´1k“1 Yk 1ptk,tk`1s in (6.83). Our aim is to

integrate more general stochastic processes. To do so the following lemma is crucial.

Lemma 6.3.22 (Density). Let T P r0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let Q P L1pUqbe nonnegative and symmetric, and let X P L0

`

PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq

˘

satisfy

that it holds P-a.s. thatşT

0Xs

2HSpQ12pUq,Hq

ds ă 8. Then there exists a sequence

Xn : r0, T s ˆΩ Ñ LpU,Hq, n P N, of pFtqtPr0,T s-elementary stochastic processes suchthat for all ε P p0,8q it holds that

limnÑ8

Pˆż T

0

Xs ´Xns

2HSpQ12pUq,Hqq ds ě ε

˙

“ 0. (6.86)

Lemma 6.3.22 follows, e.g., from Proposition 2.3.8 in Prevot & Rockner [18].In the next result, Theorem 6.3.23, the existence and uniqueness of the stochasticintegral is established (cf., e.g., Proposition 2.26 in Karatzas & Shreve [12]).


Theorem 6.3.23 (Stochastic integral*). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üqbe separable R-Hilbert spaces, let T P p0,8q, let Q P L1pUq be nonnegative and sym-metric, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, and let W : r0, T s ˆ Ω Ñ U bea Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process. Then there exists a unique lin-

ear mapping I :

X P L0`


˘

: P`

∫T0 Xs2HSpQ12pUq,Hq

ds ă

8˘

“ 1(

Ñ L0pP; ¨Hq which satisfies

(i) that for all Xn P DpIq, n P N, with lim supnÑ8E“

mint1,şT

0Xn

s 2HSpQ12pUq,Hq

dsu‰

“ 0 it holds that lim supnÑ8E“

mint1, IpXnqHu‰

“ 0 (continuity) and

(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds that

Ip1ps,tsXq “ rX pWt ´WsqsP,BpHq (6.87)

(stochastic integration of elementary stochastic processes).

Definition 6.3.24 (Stochastic integral on the entire time interval*). Let T P p0,8q,let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üqbe separable R-Hilbert spaces, let Q P L1pUq be nonnegative and symmetric, and letW : r0, T s ˆ Ω Ñ U be a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process. Then we

denote by IW :

X P L0`


˘

: P`

∫T0 Xs2HSpQ12pUq,Hq

ds ă

8˘

“ 1(

Ñ L0pP; ¨Hq the unique linear mapping which satisfies

(i) that for all Xn P DpIW q, n P N, with lim supnÑ8E“

mint1,şT

0Xn

s 2HSpQ12pUq,Hq

dsu‰

“ 0 it holds that lim supnÑ8E“

mint1, IW pXnqHu‰

“ 0 (continuity) and

(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds that

Ip1ps,tsXq “ rX pWt ´WsqsP,BpHq (6.88)

(stochastic integration of elementary stochastic processes).

Class exercise 6.3.25 (Measurability of truncated processes*). Let T P p0,8q,a P r0, T q, b P pa, T s, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , Üq be separable R-Hilbert spaces, let Q P L1pUq be nonnegativeand symmetric, let W : r0, T s ˆ Ω Ñ U be a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wienerprocess, and let

X P L0`


˘

. (6.89)

Is it true or is it not true that r0, T s ˆ Ω ÞÑ 1pa,bqpsq ¨ Xspωq P HSpQ12pUq, Hqq P

L0`


˘

?


Definition 6.3.26 (Stochastic integral on pa, bq*). Let T P p0,8q, a P r0, T q,b P pa, T s, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq andpU, 〈¨, ¨〉U , Üq be separable R-Hilbert spaces, let Q P L1pUq be nonnegative and sym-metric, let W : r0, T s ˆΩ Ñ U be a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process,

and let X P L0pPP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hqq satisfy that Pp∫ ba Xs2HSpQ12pUq,Hq

ds ă

8q “ 1. Then we denote byż b

a

Xs dWs P L0pP; ¨Hq (6.90)

the set given byşb

aXs dWs “ IW p1pa,bqXq.

Exercise 6.3.27. Let T P r0,8q, a P r0, T q, b P pa, T s, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbertspaces, let Q P L1pUq be nonnegative and symmetric, let W : r0, T s ˆ Ω Ñ U be aQ-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process, and let

X P L0`

PpP,pFtqtPr0,T sq; ¨HSpQ12pUq,Hq

˘

(6.91)

satisfy thatşb

aE“

Xs2HSpQ12pUq,Hq

‰

ds ă 8. Prove

(i) that E“

şb

aXs dWs

2H

‰

“şb

aE“

Xs2H

‰

ds ă 8 and

(ii) that E“ şb

aXs dWs

‰

“ 0.

In the following a few properties of the stochastic integral are collected. For thisthe following lemma is used.

Lemma 6.3.28 (*). Let T P r0,8q, t P r0, T s, let pΩ,F ,P, pFsqsPr0,T sq be a stochasticbasis, let pS,Sq be a measurable space, let X : Ω Ñ S be an F/S-measurable mappingand let Y : Ω Ñ S be an Ft/S-measurable mapping such that it holds P-a.s. thatX “ Y . Then it holds that X is Ft/S-measurable.

Proof of Lemma 6.3.28*. First, note that the assumption that X “ Y P-a.s. showsthat there exists a measurable set A P F with the property that PpAq “ 1 and withthe property that for all ω P A it holds that Xpωq “ Y pωq. Next observe that for allB P S it holds that

X´1pBq “

“

X´1pBq X A

‰

Y“

X´1pBqzA

‰

“ tω P A : Xpωq P Bu Y“

X´1pBqzA

‰

“ tω P A : Y pωq P Bu Y“

X´1pBqzA

‰

““

Y ´1pBq X A

‰

Y“

X´1pBqzA

‰

.

(6.92)


Moreover, observe that the assumption that pFtqtPr0,T s is a normal filtration togetherwith the fact that PpAq “ 1 implies that

A,Ac P F0 Ď Ft Ď F . (6.93)

This and the assumption that Y is Ft/S-measurable prove that for all B P S it holdsthat

Y ´1pBq X A P Ft. (6.94)

Furthemore, note that the monotonicity of the probability measure P ensures thatfor all B P S it holds that PpX´1pBqzAq “ 0. The assumption that pFtqtPr0,T s isnormal hence shows that for all B P S it holds that

X´1pBqzA P Ft. (6.95)

Combining (6.92) with (6.94) and (6.95) proves that for all B P S it holds thatX´1pBq P Ft. The proof of Lemma 6.3.28 is thus completed.

Consider the setting of Lemma 6.3.28 and let pV, ¨V q be a separable Banachspace. Then Lemma 6.3.28, in particular, proves that for all t1, t2 P r0, T s witht1 ď t2 it holds that

L0`

P|Ft1 ; ¨V˘

Ď L0`

P|Ft2 ; ¨V˘

Ď L0`

P; ¨V˘

. (6.96)

In Lemma 6.3.28 it is crucial that the filtration is normal. Let us collect a few prop-erties of the stochastic integral with possibly infinite dimensional Wiener processesas integrator processes.


Theorem 6.3.29 (Properties of the stochastic integral*). Let T P p0,8q, a P r0, T q,b P pa, T s, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbert spaces, letQ P L1pUq be nonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic ba-sis, let W : r0, T sˆΩ Ñ U be a Q-standard pΩ,F ,P, pFtqtPr0,T sq-Wiener process, andlet X : r0, T sˆΩ Ñ HSpQ12pUq, Hq be an pFtqtPr0,T s/BpHSpQ12pUq, Hqq-predictable

stochastic processes which satisfies that P` şb

aXs

2HSpQ12pUq,Hq ds ă 8

˘

“ 1. Then

(i) it holds thatşb

aXs dWs P L

0`

P|Fb ; ¨H˘

,

(ii) for all α, β P R and all pFtqtPr0,T s/BpHSpQ12pUq, Hqq-predictable stochastic

processes Y, Z : r0, T s ˆ Ω Ñ HSpQ12pUq, Hq with P` şb

aYs

2HSpQ12pUq,Hq

`

Zs2HSpQ12pUq,Hq

ds ă 8˘

“ 1 it holds that

ż b

a

rαYs ` βZss dWs “ α

ż b

a

Ys dWs ` β

ż b

a

Zs dWs, (6.97)

(iii) for all pFtqtPr0,T s/BpHSpQ12pUq, Hqq-predictable stochastic processes Y : r0, T sˆ

Ω Ñ HSpQ12pUq, Hq withşb

aE“

Ys2HSpQ12pUq,Hq

‰

ds ă 8 it holds that

E

«

›

›

›

›

ż b

a

Ys dWs

›

›

›

›

2

H

ff

“

ż b

a

E

”

Ys2HSpQ12pUq,Hq

ı

ds ă 8, (Ito’s isometry)

›

›

›

›

ż b

a

Ys dWs

›

›

›

›

L2pP;¨Hq

“

ˆż b

a

Ys2L2pP;¨

HSpQ12pUq,Hqqds

˙

12

ă 8, (6.98)

E

„ż b

a

Ys dWs

“ 0, (6.99)

(iv) for all p P r2,8q it holds that

›

›

›

›

ż b

a

Xs dWs

›

›

›

›

LppP;¨Hq

ď

c

p pp´ 1q

2

ˆż b

a

Xs2LppP;¨

HSpQ12pUq,Hqqds

˙

12

,

˜

E

«

›

›

›

›

ż b

a

Xs dWs

›

›

›

›

p

H

ff¸1p

ď

c

p pp´ 1q

2

ˆż b

a

´

E

”

Xsp

HSpQ12pUq,Hq

ı¯2pds

˙

12

,

(Burkholder-Davis-Gundy inequality I)


(v) there exists an up to indistinguishability unique pFtqtPra,bs/BpHq-adapted stochas-tic process V : ra, bs ˆ Ω Ñ H with continuous sample paths which satisfies forall t P ra, bs that rVtsP,BpHq “

şt

aXs dWs (V is called a continuous modification

of pşt

aXs dWsqtPra,bs), and

(vi) for all continuous modifications V : ra, bs ˆ Ω Ñ H of pşt

aXs dWsqtPra,bs and all

p P r2,8q it holds that

›

›

›

›

›

supsPra,bs

VsH

›

›

›

›

›

LppP;|¨|Rq

ď p

ˆż b

a

Xs2LppP;¨

HSpQ12pUq,Hqqds

˙

12

,

˜

E

«

supsPra,bs

VspH

ff¸1p

ď p

ˆż b

a

´

E

”

Xsp

HSpQ12pUq,Hq

ı¯2p

ds

˙

12

.

(Burkholder-Davis-Gundy inequality II)

The statements of Theorem 6.3.29 and their proofs can, for example, be found in[18] and [3].

Exercise 6.3.30 (Stochastic integration of L2-continuous stochastic processes). LetT P p0,8q, d,m P N, let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let W : r0, T sˆΩ ÑRm be an m-dimensional standard pFtqtPr0,T s-Brownian motion, let a, b P r0, T s witha ď b and let X : r0, T sˆΩ Ñ Rdˆm be an pFtqtPr0,T s/BpRdˆmq-predictable stochasticprocess with X P Cpr0, T s, L2pP ; ¨

Rdˆmqq. Prove then that

ż b

a

Xs dWs “ L2pP ; ¨

Rdq´ lim

nÑ8

«

n´1ÿ

k“0

Xpa` kpbáq

nq

´

Wa` pk`1qpbáq

n´W

a` kpbáqn

¯

ff

.

(6.100)

6.3.5 On the density of elementary processes

In this subsection density properties of elementary processes are investigated. Forthis the following notation is used.

Definition 6.3.31 (Round down to the grid). We denote by tüh : RÑ R, h P p0,8q,the mappings which satisfy for all h P p0,8q, t P R that

ttuh “ max`

p´8, ts X t0, h,´h, 2h,´2h, . . . u˘

. (6.101)


Definition 6.3.32 (Down to the grid). We denote by z¨h : R Ñ R, h P p0,8q, themappings which satisfy for all h P p0,8q, t P R that

zth “ max`

p´8, tq X t0, h,´h, 2h,´2h, . . . u˘

. (6.102)

The following results are based on Theorem 25.9 in Klenke [13].

Lemma 6.3.33. Let pV, ¨V q be a normed R-vector space, let a P R, b P pa,8q,let pΩ,F ,P, pFtqtPra,bsq be a filtered probability space, let X : ra, bs ˆ Ω Ñ V be apFtqtPra,bs/BpV q-adapted stochastic process with continuous sample paths which satis-fies suptPra,bs supωPΩ XtpωqV ă 8, let hn P p0,8q, n P N, satisfy lim supnÑ8 hn “ 0,and let Y n : ra, bs ˆ Ω Ñ V , n P N, be the mappings which satisfy for all n P N,t P pa, bs that Y n

a “ 0 andY nt “ Xa`ztáhn

. (6.103)

Then

(i) it holds for every n P N that Y n is a pFtqtPra,bs/BpV q-adapted stochastic processwith caglad sample paths and

(ii) it holds that

lim supnÑ8

E

„ż b

a

Xt ´ Ynt

2V dt

“ 0. (6.104)

Proof of Lemma 6.3.33. Observe that the fact that @ a, b P R : pa ` bq2 ď 2a2 ` 2b2

ensures that

suptPra,bs

supωPΩ

Xtpωq ´ Ynt pωq

2V ď 4

«

suptPra,bs

supωPΩ

Xtpωq2V

ff

ă 8. (6.105)

In the next step we combine this and the fact that

@ t P r0, T s, ω P Ω: lim supnÑ8

Xtpωq ´ Ynt pωqV “ 0 (6.106)

with Lebesgue’s theorem of dominated convergence to obtain that

lim supnÑ8

ż

ra,bsˆΩ

Xtpωq ´ Ynt pωq

2V pBorelra,bsbPqpdt, dωq “ 0. (6.107)



Lemma 6.3.34. Let pV, ¨V q be a separable R-Banach space, let a P R, b P pa,8q,let pΩ,F ,P, pFtqtPra,bsq be a filtered probability space, let hn P p0,8q, n P N, sat-isfy lim supnÑ8 hn “ 0, let X : ra, bs ˆ Ω Ñ V be a pFtqtPra,bs/BpV q-progressivelymeasurable stochastic process which satisfies suptPra,bs supωPΩ XtpωqV ă 8, and letY n : ra, bs ˆ Ω Ñ V , n P N, be the mappings which satisfy for all n P N, t P ra, bsthat

Y nt “

1

hn

ż t

maxtt´hn,au

Xs ds. (6.108)

Then

(i) it holds for every n P N that Y n is a pFtqtPra,bs/BpV q-adapted stochastic processwith continuous sample paths,

(ii) it holds for all ω P Ω that

supnPN

suptPra,bs

Y nt pωqV ď sup

tPra,bs

XtpωqV ď suptPra,bs

supwPΩ

XtpwqV ă 8, (6.109)

and

(iii) it holds for all p P p0,8q that

lim supnÑ8

E

„ż b

a

Xt ´ Ynt

pV dt

“ 0. (6.110)

Proof of Lemma 6.3.34. Throughout this proof let Z : ra, bsˆΩ Ñ V be the mappingwhich satisfies for all t P ra, bs that

Zt “

ż t

a

Xs ds. (6.111)

Next observe that for all n P N, t P ra, bs it holds that

Y nt “

1hn

`

Zt ´ Zmaxtt´hn,au

˘

. (6.112)

This together with the assumption that X is pFtqtPra,bs/BpV q-progressively measur-able establishes item (i). Moreover, note that item (ii) follows immediately from theassumption that suptPra,bs supωPΩ XtpωqV ă 8. It thus remains to prove item (iii).For this observe that (6.112) ensures that for all ω P Ω, r P pa, bq it holds that

Borelrr,bs

ˆ"

t P rr, bs : lim supnÑ8

Xtpωq ´ Ynt pωqV “ 0

*˙

“ b´ r. (6.113)


This, in turn, implies that for all ω P Ω, p P p0,8q it holds that

Borelra,bs

ˆ"

t P ra, bs : lim supnÑ8

Xtpωq ´ Ynt pωq

pV “ 0

*˙

“ b´ a. (6.114)

Lebesgue’s theorem of dominated convergence and item (ii) hence imply that for allω P Ω, p P p0,8q it holds that

lim supnÑ8

ż b

a

Xtpωq ´ Ynt pωq

pV dt “ 0. (6.115)

This, again Lebesgue’s theorem of dominated convergence, and again item (ii) showthat for all p P p0,8q it holds that

lim supnÑ8

ż

Ω

ż b

a

Xtpωq ´ Ynt pωq

pV dtPpdωq “ 0. (6.116)


Lemma 6.3.35. Let pV, ¨V q be a separable R-Banach space, let p P p0,8q, a P R,b P pa,8q, let pΩ,F ,P, pFtqtPra,bsq be a filtered probability space, let X : ra, bsˆΩ Ñ Vbe a pFtqtPra,bs/BpV q-progressively measurable stochastic process, and let Y n : ra, bs ˆΩ Ñ V , n P N, be the mappings which satisfy for all n P N, t P ra, bs that

Y nt “ Xt1twPΩ: XtpwqV ănupωq. (6.117)

Then

(i) it holds for every n P N that Y n is a pFtqtPra,bs/BpV q-progressively measurablestochastic process,

(ii) it holds for all n P N that

suptPra,bs

supωPΩ

Y nt pωqV ď n (6.118)

and

(iii) it holds for all ω P Ω withşb

aXtpωq

pV dt ă 8 that

lim supnÑ8

ż T

0

Xtpωq ´ Ynt pωq

pV dt “ 0. (6.119)


Proof of Lemma 6.3.35. Throughout this proof let Zn : ra, bs ˆ Ω Ñ R, n P N, bethe mappings which satisfies for all n P N, t P ra, bs, ω P Ω that

Znt pωq “ 1twPΩ: XtpwqV ănupωq. (6.120)

Observe that the assumption that X is pFtqtPra,bs/BpV q-measurable ensures that forall n P N, τ P ra, bs, A P PpRzt0uq with 1 P A it holds that

tpt, ωq P ra, τ s ˆ Ω: Znt pωq P Au “ tpt, ωq P ra, τ s ˆ Ω: Zn

t pωq “ 1u

“ tpt, ωq P ra, τ s ˆ Ω: Xnt pωqV ă nu P pBpra, τ sq b Fτ q .

(6.121)

This implies that for every n P N it holds that Zn is a pFtqtPra,bs/BpRq-progressivelymeasurable stochastic process. This establishes item (i). Item (ii) is an immediateconsequence from (6.117). It thus remains to prove item (iii). For this observe that

Lebesgue’s theorem of dominated convergence ensures that for all ω P tşb

aXt

p dt ă8u it holds that

lim supnÑ8

ż T

0

Xtpωq ´ Ynt pωq

pV dt

“ lim supnÑ8

ż T

0

XtpωqpV 1twPΩ: XtpwqV ěnupωq dt “ 0.

(6.122)


Corollary 6.3.36. Let pV, ¨V q be a separable R-Banach space, let p, c P p0,8q,a P R, b P pa,8q, let pΩ,F ,P, pFtqtPra,bsq be a filtered probability space, and letX : ra, bs ˆ Ω Ñ V be a pFtqtPra,bs/BpV q-progressively measurable stochastic process

with P` şb

aXt

pV dt ă 8

˘

“ 1. Then there exists a sequence Y n : ra, bs ˆ Ω Ñ V ,n P N, of pFtqtPra,bs/BpV q-predictable stochastic processes

(i) which satisfies that for all n P N it holds that suptPra,bs supωPΩ Ynt pωqV ă 8,

(ii) which satisfies that for all n P N there exists a h P p0,8q such that for allω P Ω, t1, t2 P pa, bs with zt1 ´ ah “ zt2 ´ ah it holds that Y n

a pωq “ 0 andY nt1pωq “ Y n

t2pωq, and

(iii) which satisfies that

lim supnÑ8

E

„

min

"

c,

ż T

0

Xtpωq ´ Ynt pωq

pV dt

*

“ 0. (6.123)

Proof of Corollary 6.3.36. Observe that items (i)–(iii) are an immediate consequencefrom Lemma 6.3.33, Lemma 6.3.34, and Lemma 6.3.35. The proof of Corollary 6.3.36is thus completed.


6.3.6 Elementary processes revisited

In the literature (see, e.g., Definition 2.3.1 in [18]) a slightly different notion ofan elementary stochastic process is often given. In Proposition 6.3.40 below weshow that these different definitions (cf. Definition 2.3.1 in [18] and Definition 6.3.16above) are equivalent. Our proof of Proposition 6.3.40 uses Exercise 6.3.38 below.Exercise 6.3.38 below can, e.g., be proved by using the following lemma.

Lemma 6.3.37. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space andlet ϕ P CpH,Kq, v P Hzt0u with the property that for all u P H it holds thatϕpuq ¨ 〈v, u〉H “ 0. Then it holds for all u P H that ϕpuq “ 0.

Proof of Lemma 6.3.37. Observe that for all λ P Kzt0u, w P rspantvusK it holds that

0 “ ϕpλv ` wq ¨ 〈v, λv ` w〉H “ ϕpλv ` wq ¨ λ ¨ v2Hlooomooon

‰0

. (6.124)

This implies that for all for all λ P Kzt0u, w P rspantvusK it holds that

ϕpλv ` wq “ 0. (6.125)

The fact that the set!

λv ` w P H : λ P Kzt0u, w P rspantvusK)

(6.126)

is dense in H together with the assumption that ϕ is continuous hence implies that forall u P H it holds that ϕpuq “ 0. The proof of Lemma 6.3.37 is thus completed.

Exercise 6.3.38. Let K P tR,Cu, n P t2, 3, . . . u, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2ube K-Hilbert spaces and let A1, . . . , An P LpH1, H2q with the property that for all u PH1 it holds that A1u P tA2u,A3u, . . . , Anuu. Prove then that A1 P tA2, A3, . . . , Anu.

Exercise 6.3.38 can, e.g., be proved by using Lemma 6.3.37. In our proof ofProposition 6.3.40 below we also use the following exercise.

Exercise 6.3.39. Let pΩ1,F1q be a measurable space, let Ω2 be a set and let f : Ω1 Ñ

Ω2 be a mapping with the property that the set impfq is finite. Then f is F1/PpΩ2q-measurable if and only if for all ω P impfq it holds that f´1ptωuq P F1.

Proposition 6.3.40 (Uniform and strong measurability). Let K P tR,Cu, let pΩ,Fqbe a measurable space, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces and letY : Ω Ñ LpH1, H2q be a function with the property that impY q is a finite set. Thenit holds that Y is F/BpLpH1, H2qq-measurable if and only if for all v P H1 it holdsthat Ω Q ω ÞÑ Y pωqv P H2 is F/BpH2q-measurable.


Proof of Proposition 6.3.40. It is clear that if Y is F/BpLpH1, H2qq-measurable, thenit holds for all v P H1 that Y v is F/BpH2q-measurable. We thus assume in thefollowing that for all v P H1 it holds that Y v is F/BpH2q-measurable. Let A1 P impY qbe arbitrary. We now prove that Y ´1ptA1uq P F . This and Exercise 6.3.39 will thenshow that Y is F/BpLpH1, H2qq-measurable. W.l.o.g. we assume that impY qztA1u ‰

H. As impY q is a finite set, there exists n P t2, 3, . . . u and A2, . . . , An P impY qztA1u

such thattA1, A2, . . . , Anu “ impY q. (6.127)

Exercise 6.3.38 implies that there exists a vector u P H1 such that

A1u R tA2u, . . . , Anuu . (6.128)

This and the assumption that Y u is F/BpH2q-measurable imply that

F Q pY uq´1ptA1uuq “ Y ´1

ptA1uq. (6.129)

Exercise 6.3.39 thus completes the proof of Proposition 6.3.40.

Exercise 6.3.41. Let pΩ,Fq be a measurable space, let pE, dEq be a metric spaceand let f : Ω Ñ E be a mapping with the property that the set impfq is finite. Thenf is F/PpEq-measurable if and only if f is F/BpEq-measurable.

6.3.7 Cylindrical Wiener process

The following presentations are based on [18] and Chapter 5 in [10]. Let pU, 〈¨, ¨〉U , Üqbe an R-Hilbert space and let Q P L1pUq be nonnegative and symmetric. In Subsec-tion 6.1.2 the notion of a Q-standard Wiener process is presented. The covarianceoperator Q associated to a Q-standard Wiener process is a nonnegative, symmetric,and nuclear linear operator on the Hilbert space U on which the standard Wienerprocess takes values in. In many situations one is interested in an infinite dimen-sional Wiener process with a covariance operator that is a nonnegative and symmetricbounded linear operator which is not a nuclear linear operator (such as, for example,the identity operator on an infinite dimensional Hilbert space). This can be achieved


by the concept of a cylindrical Wiener process.

Definition 6.3.42 (Cylindrical Wiener process*). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hqand pH1, 〈¨, ¨〉H1

, ¨H1q be R-Hilbert spaces with H Ď H1 continuously, let Q P LpHq

and Q1 P L1pH1q be nonnegative and symmetric with the property that

`

Q12pHq,

›

›Q´12p¨q›

›

H

˘

“`

Q12

1 pH1q,›

›Q´12

1 p¨q›

›

H1

˘

, (6.130)

and let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space. Then we say that W is aQ-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process if and only if W is a Q1-standardpΩ,F ,P, pFtqtPr0,T sq-Wiener process.

Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let Q P LpHq benonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space,and let pWtqtPr0,T s be a Q-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process. The Q-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process pWtqtPr0,T s thus, in general, does nottake values in the Hilbert space H, on which the covariance operator Q associatedto W is defined, but on a larger Hilbert space with a weaker topology into which His continuously embedded. More results on Q-cylindrical Wiener processes can befound in Section 2.5 in Prevot & Rockner [18].


Part III

Stochastic Partial DifferentialEquations (SPDEs)

223

Chapter 7

Solutions of SPDEs

7.1 Existence, uniqueness and properties of mild

solutions of SPDEs

7.1.1 Mild solutions of SPDEs

The next definition presents what we mean by a stochastic partial differential equa-tion and a pΩ,F ,P, pFtqtPr0,T sq-mild solution of it.

225

226 CHAPTER 7. SOLUTIONS OF SPDES

Definition 7.1.1 (Mild solutions*). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be sep-arable R-Hilbert spaces, let Q P LpUq be nonnegative and symmetric, let A : DpAq ĎH Ñ H be a diagonal linear operator with suppσP pAqq ă 8, let η P psuppσP pAqq,8q,T P r0,8q, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associ-ated to ηÁ, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let α, β, γ P R, O P BpHγq,B PM

`

BpOq,BpHSpQ12pUq, Hβqq˘

, F PM`

BpOq,BpHαq˘

, ξ PMpF0,BpOqq, andlet pWtqtPr0,T s be a Q-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process. Then we saythat X is an pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (7.1)

if and only if it holds that X PMpr0, T sˆΩ, Oq is a pFtqtPr0,T s/BpHSpQ12pUq, Hγqq-predictable stochastic process which satisfies for all t P r0, T s that

Pˆż t

0

eApt´sqF pXsqHγ ` eApt´sqBpXsq

2HSpQ12pUq,Hγq

ds ă 8

˙

“ 1 (7.2)

and

rXtsP,BpHγq “

„

eAtξ `

ż t

0

1tşt0 e

AptúqF pXuqHγ duă8ueApt´sqF pXsq ds

P,BpHγq

`

ż t

0

eApt´sqBpXsq dWs.

(7.3)

Remark 7.1.2 (*). Equation (7.1) is referred to as stochastic partial differentialequation (SPDE), the function F is the nonlinear part of the drift coefficient functionAv ` F pvq, v P O, and the function B is the diffusion coefficient function of theSPDE (7.1).

7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 227

Example 7.1.3 (Ornstein-Uhlenbeck processes (stochastic heat equation with ad-ditive noise)). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplaceoperator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq (see Definition 3.6.10),let ξ P L2pBorelp0,1q; |¨|Rq, let en P L2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all

n P N and Borelp0,1q-almost all x P p0, 1q it holds that enpxq “?

2 sinpnπxq, letpΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let rn P r0,8q, n P N, be a bounded se-quence of nonnegative real numbers, let Q : L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rqbe the mapping with the property that for all v P L2pBorelp0,1q; |¨|Rq it holds that

Qv “8ÿ

n“1

rn 〈en, v〉L2pBorelp0,1q;|¨|Rqen, (7.4)

let pWtqtPr0,T s be a Q-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, and let X : r0, T sˆΩ Ñ H be an pFtqtPr0,T s/BpHq-predictable stochastic process which fulfills that for allt P r0, T s it holds P-a.s. that

Xt “ eAtξ `

ż t

0

eApt´sq dWs. (7.5)

Then X is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXt “ AXt dt` dWt, t P r0, T s, X0 “ ξ. (7.6)

Sometimes one also writes

dXtpxq “B2

Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq (7.7)

for t P r0, T s, x P p0, 1q as a short form for (7.6).

In the following we investigate a few further properties of mild solutions of SPDEs.To this end we frequently use the following setting.

7.1.2 A setting for SPDEs with globally Lipschitz continuousnonlinearities*

Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be two separable R-Hilbert spaces, letA : DpAq Ď H Ñ H be a diagonal linear operator with suppσP pAqq ă 0, letpHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to Á,let T P r0,8q, p P r2,8q, γ P R, η P r0, 1q, β P rγ ´ η2, γs, F P C0,1pHγ, Hγ´ηq,B P C0,1pHγ, HSpU,Hβqq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let ξ PLppP|F0 ; ¨Hγ q, and let pWtqtPr0,T s be an IdU -cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wienerprocess.


7.1.3 A strong perturbation estimate for SPDEs

The following estimate is a special case of an inequality known as Minkowski’s integralinequality; see, for instance, Apendix A.1 in Stein [20] and Theorem 202 in Hardy,Littlewood & Polya [4].

Proposition 7.1.4 (*). Let T P r0,8q, p P r1,8q, let pΩ,F ,Pq be a probabilityspace, and let Y : r0, T s ˆ Ω Ñ r0,8s be an pBpr0, T sq b Fq/Bpr0,8sq-measurablemapping. Then

ˆ

E„ˇ

ˇ

ˇ

ˇ

ż T

0

Ys ds

ˇ

ˇ

ˇ

ˇ

p˙1p

ď

ż T

0

´

E“

|Ys|p‰

¯1pds. (7.8)

Definition 7.1.5 (Fat integral*). Let pΩ,F ,Pq be a probability space, let a P R,b P ra,8q, let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let X : ra, bs ˆΩ Ñ H

be an pBpra, bsqbFq/BpHq-measurable mapping. Then we denote byşb

aXs ds the set

given by

ż b

a

Xs ds “

„ż b

a

1tşba |Xr| dră8u

Xs ds

P,BpHq

“

„

Ω Q ω ÞÑ

ż b

a

1twPΩ:

şba |Xrpwq| dră8u

pωqXspωq ds P H

P,BpHq.

(7.9)

In the next result, Proposition 7.1.6, we establish, in the setting of Section 7.1.2,a certain strong perturbation result for two arbitrary predictable stochastic processesX1 and X2 satisfying supsPr0,T s maxkPt1,2u X

ks LppP;¨Hγ q

ă 8. In particular, we em-

phasize in the setting of Section 7.1.2 that neither X1 nor X2 in Proposition 7.1.6need to be pΩ,F ,P, pFtqtPr0,T sq-mild solutions of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.10)

In the statement of Proposition 7.1.6 and in a number of other results in this andthe next chapter we use the functions Er : r0,8q Ñ r0,8q, r P p0,8q, introduced inDefinition 1.3.1 in Chapter 1 above.


Proposition 7.1.6 (Perturbation estimate*). Assume the setting in Section 7.1.2and let X1, X2 : r0, T sˆΩ Ñ Hγ be pFtqtPr0,T s/BpHγq-predictable stochastic processeswith the property that supsPr0,T s maxkPt1,2u X

ks LppP;¨Hγ q

ă 8. Then

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?

2 |F |C0,1pHγ,Hγ´ηq?1´η

à

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq

¨?

2 suptPr0,T s

›

›

›

›

„

“

X1t

‰

P,BpHγq´

ż t

0

eApt´sqF pX1s qds´

ż t

0

eApt´sqBpX1s q dWs

´

„

“

X2t

‰

P,BpHγq´

ż t

0

eApt´sqF pX2s qds´

ż t

0

eApt´sqBpX2s q dWs

›

›

›

›

LppP;¨Hγ q

ă 8.

(7.11)

Proof of Proposition 7.1.6*. Note that the Minkowski inequality ensures that for allt P r0, T s it holds that

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď

›

›

›

›

›

„

“

X1t

‰

P,BpHγq´

ˆż t

0

eApt´sqF pX1s qds`

ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0

eApt´sqF pX2s qds`

ż t

0

eApt´sqBpX2s q dWs

˙

´“

X2t

‰

P,BpHγq

›

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

›

ˆż t

0

eApt´sqF pX1s qds`

ż t

0

eApt´sqBpX1s q dWs

˙

´

ˆż t

0

eApt´sqF pX2s qds`

ż t

0

eApt´sqBpX2s q dWs

˙

›

›

›

›

›

LppP;¨Hγ q

.

(7.12)


Again the Minkowski inequality hence implies that for all t P r0, T s it holds that

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď

›

›

›

›

›

„

“

X1t

‰

P,BpHγq´

ˆż t

0

eApt´sqF pX1s qds`

ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0

eApt´sqF pX2s qds`

ż t

0

eApt´sqBpX2s q dWs

˙

´“

X12

‰

P,BpHγq

›

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

.

(7.13)

Next note that Holder’s inequality implies that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

ď

ż t

0

›

›eApt´sq“

F pX1s q ´ F pX

2s q‰›

›

LppP;¨Hγ qds

ď

ż t

0

F pX1s q ´ F pX

2s qLppP;¨Hγ´η

q

pt´ sqηds

ď |F |C0,1pHγ ,Hγ´ηq

ż t

0

X1s ´X

2s LppP;¨Hγ q

pt´ sqηds


g

f

f

e

ż t

0

pt´ sq´η ds

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds

“ |F |C0,1pHγ ,Hγ´ηq

d

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η X1s ´X

2s

2LppP;¨Hγ q

ds.

(7.14)


Furthermore, observe that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

ď

d

p pp´1q2

ż t

0

eApt´sq rBpX1s q ´BpX

2s qs

2LppP;¨HSpU,Hγ qq

ds

ď

d

p pp´1q2

ż t

0

pt´ sq´η BpX1s q ´BpX

2s q

2LppP;¨HSpU,Hγ´η2q

qds

ď |B|C0,1pHγ ,HSpU,Hγ´η2qq

d

p pp´1q2

ż t

0

pt´ sq´η X1s ´X

2s

2LppP;¨Hγ q

ds.

(7.15)

Combining (7.14) and (7.15) proves that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

ď

„

|F |C0,1pHγ ,Hγ´ηqtp1´ηq2?

1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq

?p pp´1q?

2

¨

g

f

f

e

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds.

(7.16)

Putting this into (7.13) and using the fact that @ a, b P R : pa`bq2 ď 2a2`2b2 proves


that for all t P r0, T s it holds that

›

›X1t ´X

2t

›

›

2

LppP;¨Hγ q

ď 2

›

›

›

›

›

„

“

X1t

‰

P,BpHγq´

ˆż t

0

eApt´sqF pX1s qds`

ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0

eApt´sqF pX2s qds`

ż t

0

eApt´sqBpX2s q dWs

˙

´“

X2t

‰

P,BpHγq

›

›

›

›

›

2

LppP;¨Hγ q

`

”

|F |C0,1pHγ ,Hγ´ηq

?2T

p1´ηq2?

1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq

a

p pp´ 1qı2

¨

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds.

(7.17)

The generalized Gronwall inequality in Corollary 1.4.6 hence completes the proof ofProposition 7.1.6.

The next corollary, Corollary 7.1.7, is an immediate consequence of the strongperturbation estimate in Proposition 7.1.6 above.

Corollary 7.1.7 (Perturbation in the initial value*). Assume the setting in Sec-tion 7.1.2 and let X1, X2 : r0, T sˆΩ Ñ Hγ be pFtqtPr0,T s/BpHγq-predictable stochasticprocesses which satisfy for all t P r0, T s, k P t1, 2u that supsPr0,T s X

ks LppP;¨Hγ q

ă 8

and

“

Xkt

‰

P,BpHγq““

eAtXk0

‰

P,BpHγq`

ż t

0

eApt´sqF pXks qds`

ż t

0

eApt´sqBpXks q dWs. (7.18)

Then

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ qď?

2›

›X10 ´X

20

›

›

LppP;¨Hγ q

¨ Ep1´ηq„

T 1´η?


à


ă 8.(7.19)


7.1.4 Uniqueness of mild solutions of SPDEs

7.1.4.1 Uniqueness of predictable mild solutions of SEEs with globallyLipschitz continuous coefficients

As an immediate consequence of Corollary 7.1.7, we obtain, under suitable assump-tions, uniqueness of mild solutions of SPDEs; cf., e.g., Theorem 7.4 (i) in Da Prato& Zabczyk [3].

Corollary 7.1.8 (*). Assume the setting in Subsection 7.1.2, let X1, X2 : r0, T s ˆΩ Ñ Hγ be pΩ,F ,P, pFtqtPr0,T sq-mild solutions of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (7.20)

and assume that maxkPt1,2u suptPr0,T s Xkt LppP;¨Hγ q

ă 8. Then it holds for all t P

r0, T s that P`

X1t “ X2

t

˘

“ 1.

7.1.4.2 Uniqueness of left-continuous mild solutions of SEEs with semi-globally Lipschitz continuous coefficients

The proof of the next result, Proposition 7.1.9, is similiar to the proof of Theorem 7.4in Da Prato & Zabczyk [3]. See also, e.g., Lemma 8.2 in [22] for the next result.

Proposition 7.1.9 (Local solutions). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq beseparable R-Hilbert spaces, let A : DpAq Ď H Ñ H be a symmetric diagonal linearoperator with suppσP pAqq ă 0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpo-lation spaces associated to Á, let T P p0,8q, γ P R, η P r0, 1q, F P CpHγ, Hγ´ηq,B P CpHγ, HSpU,Hγ´η2qq satisfy for all bounded sets E Ď Hγ that |F |E|C0,1pE,Hγ´ηq`

|B|E|C0,1pE,HSpU,Hγ´η2qq ă 8, let pΩ,F ,Pq be a probability space with a normal fil-tration pFtqtPr0,T s, let τk : Ω Ñ r0, T s, k P t1, 2u, be pFtqtPr0,T s-stopping times, andlet Xk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s/BpHγq-adapted stochastic pro-cesses with left-continuous and bounded sample paths which satisfy for all k P t1, 2u,t P r0, T s that P

` şt

01tsăτku

“

ept´sqAF pXks qHγ ` e

pt´sqABpXks q

2HSpU,Hγq

‰

ds ă 8˘

“ 1and

“

Xkt 1ttďτku

‰

P,BpHγq(7.21)

“

„

“

etAXk0

‰

P,BpHγq`

ż t

0

1tsăτku ept´sqAF pXk

s qds`

ż t

0

1tsăτku ept´sqABpXk

s q dWs

1ttďτku.

Then P`

@ t P r0, T s : 1tX10“X

20uX1

mintt,τ1,τ2u“ 1tX1

0“X20uX2

mintt,τ1,τ2u

˘

“ 1.


Proof of Proposition 7.1.9. Throughout this proof let Ω P F be the set given byΩ “ tX0 “ Y0u, let %r,k : Ω Ñ r0, T s, r P p0,8q, k P t1, 2u, be the mappings with theproperty that for all r P p0,8q, k P t1, 2u it holds that

%r,k “ inf`

tT u Y tt P r0, T s : Xkt Hγ ą ru

˘

, (7.22)

let ρr : Ω Ñ r0, T s, r P p0,8q, be the mappings with the property that for allr P p0,8q it holds that ρr “ mint%r,1, %r,2, τ1, τ2u, and let Xk,r : r0, T s ˆ Ω Ñ Hγ,k P t1, 2u, r P p0,8q, be the mappings with the property that for all k P t1, 2u,r P p0,8q, t P r0, T s it holds that Xk,r

t “ 1ΩXttďρruXkt . Note for all r P p0,8q, k P N

that %r,k and ρr are pFtqtPr0,T s-stopping times. This ensures that for every r P p0,8q,k P N it holds that Xk,r is a pFtqtPr0,T s/BpHγq-predictable stochastic process withleft-continuous sample paths. Moreover, observe that for all r P p0,8q, t P r0, T s itholds P-a.s. that

X1,rt ´X2,r

t “ 1ΩXttďρru

ż t

0

ept´sqA“

1tsăτ1uF pX1s q ´ 1tsăτ2uF pX

2s q‰

ds

` 1ΩXttďρru

ż t

0

ept´sqA“

1tsăτ1uBpX1s q ´ 1tsăτ2uBpX

2s q‰

dWs

“ 1ΩXttďρru

ż t

0

1ΩXtsăρru ept´sqA

“

F pX1s q ´ F pX

2s q‰

ds

` 1ΩXttďρru

ż t

0


“

BpX1s q ´BpX

2s q‰

dWs

“ 1ΩXttďρru

ż t

0


“

F pX1,rs q ´ F pX

2,rs q

‰

ds

` 1ΩXttďρru

ż t

0


“

BpX1,rs q ´BpX

2,rs q

‰

dWs.

(7.23)


This implies for all r P p0,8q that

supsPr0,T s

E“

X1,rs ´X2,r

s 2Hγ

‰

“ supsPr0,T s

E“

1ΩXtsďρruX1s ´X

2s

2Hγ

‰

ď supsPr0,T s

E“

1ΩXtsďρruXtρr“0uX1s ´X

2s

2Hγ

‰

` supsPr0,T s

E“

1ΩXtsďρruXtρrą0uX1s ´X

2s

2Hγ

‰

“ supsPr0,T s

E“

1ΩXtsďρruXtρrą0uX1s ´X

2s

2Hγ

‰

ď supsPr0,T s

E“

1ΩXtρrą0uX1mints,ρru ´X

2mints,ρru

2Hγ

‰

ď 2 ¨ supsPr0,T s

E“

1tρrą0uX1mints,ρru

2Hγ ` 1tρrą0uX

2mints,ρru

2Hγ

‰

ď 4r2ă 8.

(7.24)

Moreover, equation (7.23) ensures that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2E

«

ˇ

ˇ

ˇ

ˇ

ż t

0

1ΩXtsăρru›

›ept´sqA“

F pX1,rs q ´ F pX

2,rs q

‰›

›

Hγds

ˇ

ˇ

ˇ

ˇ

2ff

` 2E

«

›

›

›

›

ż t

0


“

BpX1,rs q ´BpX

2,rs q

‰

dWs

›

›

›

›

2

Hγ

ff

.

(7.25)

Ito’s isometry hence proves that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2E

«

ˇ

ˇ

ˇ

ˇ

ż t

0

1ΩXtsăρru pt´ sq´ηF pX1,r

s q ´ F pX2,rs qHγ´η ds

ˇ

ˇ

ˇ

ˇ

2ff

` 2

ż t

0

pt´ sq´η E“

1ΩXtsăρru BpX1,rs q ´BpX

2,rs q

2HSpU,Hγ´η2q

‰

ds.

(7.26)

This shows that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2

ˆż t

0

pt´ sq´η ds

˙ż t

0

pt´ sq´η E“

1tsăρru F pX1,rs q ´ F pX

2,rs q

2Hγ´η

‰

ds

` 2

ż t

0

pt´ sq´η E“

1tsăρru BpX1,rs q ´BpX

2,rs q

2HSpU,Hγ´η2q

‰

ds.

(7.27)


Hence, we obtain that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď2T p1´ηq

p1´ ηq

ż t

0

|F |txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,Hγ´ηq

pt´ sqηE“

X1,rs ´X2,r

s 2Hγ

‰

ds

` 2

ż t

0

|B|txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,HSpU,Hγ´η2qq

pt´ sqηE“

X1,rs ´X2,r

s 2Hγ

‰

ds.

(7.28)

Combining this with (7.24) allows us to apply Corollary 1.4.6 to obtain that for allt P r0, T s, r P p0,8q it holds that E

“

X1,rt ´ X2,r

t 2Hγ

‰

“ 0. Monotone convergencehence proves that for all t P r0, T s it holds that

E“

1ttďmintτ1,τ2uuX1t ´X

2t

2Hγ

‰

“ E

”

limrÑ8

1ttďρruX1t ´X

2t

2Hγ

ı

“ limrÑ8

E“

1ttďρruX1t ´X

2t

2Hγ

‰

“ limrÑ8

E“

X1,rt ´X2,r

t 2Hγ

‰

“ 0.(7.29)

This proves that for all t P r0, T s it holds P-a.s. that 1ttďmintτ1,τ2uu rX1t ´ X2

t s “ 0.Combining this with the fact that for every ω P Ω it holds that the function r0, T s Qt ÞÑ 1ttďmintτ1pωq,τ2pωquu rX

1t pωq ´X

2t pωqs P Hγ is left-continuous ensures that

P`

@ t P r0, T s : X1mintt,τ1,τ2u

“ X2mintt,τ1,τ2u

˘

“ P`

@ t P r0, T s : 1ttďmintτ1,τ2uu rX1t ´X

2t s “ 0

˘

“ P`

@ t P r0, T s XQ : 1ttďmintτ1,τ2uu rX1t ´X

2t s “ 0

˘

“ 1.

(7.30)


Corollary 7.1.10 is an immediate consequence from Proposition 7.1.9.

Corollary 7.1.10. Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbertspaces, let T P p0,8q, γ P R, η P r0, 1q, F P CpHγ, Hγ´ηq, B P CpHγ, HSpU,Hγ´η2qq

satisfy for all bounded sets E Ď Hγ that |F |E|C0,1pE,Hγ´ηq ` |B|E|C0,1pE,HSpU,Hγ´η2qq ă

8, let pΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, and letXk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s/BpHγq-adapted stochastic processeswith continuous sample paths such that for all k P t1, 2u, t P r0, T s it holds P-a.s.that

Xkt “ etAX1

0 `

ż t

0

ept´sqAF pXks q ds`

ż t

0

ept´sqABpXks q dWs. (7.31)

Then P`

@ t P r0, T s : X1t “ X2

t

˘

“ 1.


7.1.5 Existence and regularity of mild solutions of SPDEs

Theorem 7.1.11 (*). Assume the setting in Subsection 7.1.2. Then there ex-ists an up to modifications unique pFtqtPr0,T s/BpHγq-predictable stochastic processX : r0, T s ˆ Ω Ñ Hγ which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is anpΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


Theorem 7.1.11 can be proved by a standard fixed point argument; see, e.g.,Chapter 5 in [10].

7.1.6 A priori bounds for mild solutions of SPDEs

7.1.6.1 A priori bounds

Definition 7.1.12 (*). Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normedK-vector spaces. Then we denote by ¨LippV,W q : MpV,W q Ñ r0,8s the mapping

which satisfies for all f PMpV,W q that

fLippV,W q “ fp0qW ` |f |C0,1pV,W q . (7.33)

Proposition 7.1.13 (A priori bounds*). Assume the setting in Subsection 7.1.2and let X : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s/BpHγq-predictable stochastic processwhich satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a pΩ,F ,P, pFtqtPr0,T sq-mildsolution of the SPDE


Then

suptPr0,T s

XtLppP;¨Hγ qď?

2 max

1, ξLppP;¨Hγ q

(

¨ Ep1´ηq„

T 1´η?

2 F LippHγ,Hγ´ηq?1´η

à

T 1´ηppp´ 1q BLippHγ ,HSpU,Hγ´η2qq

ă 8.(7.35)


Proof of Proposition 7.1.13*. Observe that Theorem 6.3.29 implies that for all t Pr0, T s it holds that

XtLppP;¨Hγ qď X0LppP;¨Hγ q

`

ż t

0

›

›eApt´sq F pXsq›

›

LppP;¨Hγ qds

`

c

p pp´ 1q

2

„ż t

0

›

›eApt´sqBpXsq›

›

2

LppP;¨HSpU,Hγ qqds

12

ď X0LppP;¨Hγ q`

„

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η F pXsq2LppP;¨Hγ´η

qds

12

`

c

p pp´ 1q

2

„ż t

0

pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q

qds

12

.

(7.36)

This shows that for all t P r0, T s it holds that

XtLppP;¨Hγ qď X0LppP;¨Hγ q

`

„ż t

0

pt´ sq´η max!

1, Xs2LppP;¨Hγ q

)

ds

12

¨

«

F LippHγ ,Hγ´ηq

d

T p1´ηq

p1´ ηq` BLippHγ ,HSpU,Hγ´η2qqq

c

p pp´ 1q

2

ff

.

(7.37)

This proves that for all t P r0, T s it holds that

max!

1, Xt2LppP;¨Hγ q

)

ď 2 max!

1, X02LppP;¨Hγ q

)

`

ż t

0

pt´ sq´η max!

1, Xs2LppP;¨Hγ q

)

ds

¨

«

F LippHγ ,Hγ´ηq

d

2T p1´ηq

p1´ ηq` BLippHγ ,HSpU,Hγ´η2qq

a

p pp´ 1q

ff2

.

(7.38)

An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.13.

7.1.6.2 A priori bounds revisited

Definition 7.1.14 (*). Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normedK-vector spaces. Then we denote by ¨LGpV,W q : MpV,W q Ñ r0,8s the mapping with

the property that for all f PMpV,W q it holds that

fLGpV,W q “ supvPV

„

fpvqWmaxt1, vV u

. (7.39)


Proposition 7.1.15 (A priori bounds revisited*). Assume the setting in Subsec-tion 7.1.2 and let X : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s/BpHγq-predictable stochasticprocess which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


Then

suptPr0,T s

›

›max

1, XtHγ

(›

›

LppP;|¨|Rqď?

2›

›max

1, ξHγ(›

›

LppP;|¨|Rq

¨ Ep1´ηq„

T 1´η?

2 F LGpHγ,Hγ´ηq?1´η

à

T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq

ă 8.(7.41)

Proof of Proposition 7.1.15. Observe that Theorem 6.3.29 implies that for all t Pr0, T s it holds that

›

›max

1, XtHγ

(›

›

LppP;|¨|Rq

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|Rq`

ż t

0

›

›eApt´sq F pXsq›

›

LppP;¨Hγ qds

`

c

p pp´ 1q

2

„ż t

0

›

›eApt´sqBpXsq›

›

2


12

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|Rq`

„

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η F pXsq2LppP;¨Hγ´η

qds

12

`

c

p pp´ 1q

2

„ż t

0

pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q

qds

12

.

(7.42)

This shows that for all t P r0, T s it holds that

›

›max

1, XtHγ

(›

›

LppP;|¨|Rq

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|Rq`

„ż t

0

pt´ sq´η›

›max

1, XsHγ

(›

›

2

LppP;|¨|Rqds

12

¨

«

F LGpHγ ,Hγ´ηq

d

T p1´ηq

p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qqq

c

p pp´ 1q

2

ff

.

(7.43)



›

›max

1, XtHγ

(›

›

2

LppP;|¨|Rqď 2

›

›max

1, X0Hγ

(›

›

2

LppP;|¨|Rq

`

ż t

0

pt´ sq´η›

›max

1, XsHγ

(›

›

2

LppP;|¨|Rqds

¨

«

F LGpHγ ,Hγ´ηq

d

2T p1´ηq

p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qq

a

p pp´ 1q

ff2

.

(7.44)

An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.15.

7.1.6.3 Strengthened regularity

Proposition 7.1.16 (Solution process of the SPDE enjoys more regularity than theintial value). Assume the setting in Subsection 7.1.2 and let X : r0, T sˆΩ Ñ Hγ be anpFtqtPr0,T s/BpHγq-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q

ă

8 and which is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


Then it holds for all t P r0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Xt ´ eAtX0 P

Hr

˘

“ 1 and

›

›Xt ´ eAtX0

›

›

LppP;¨Hr qď max

!

1, supsPr0,T s XsLppP;¨Hγ q

)

¨

«

F LippHγ ,Hγ´ηqtp1`γ´η´rq

p1` γ ´ η ´ rq`

a

p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq

?2 p1` 2β ´ 2rq12

ff

ă 8

(7.46)

and it holds for all t P p0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Xt P Hr

˘

“ 1and

XtLppP;¨Hr qď

X0LppP;¨Hγ q

tpr´γq`max

!

1, supsPr0,T s XsLppP;¨Hγ q

)

¨

„

F LippHγ,Hγ´ηqtp1`γ´η´rq

p1`γ´η´rq`

?p pp´1q BLippHγ,HSpU,Hβqq

tp12`β´rq

?2 p1`2β´2rq12

ă 8.(7.47)


Proof of Proposition 7.1.16. First of all, recall that for all t P r0, T s it holds that

“

Xt ´ eAtX0

‰

P,BpHγq“

ż t

0

eApt´sqF pXsqds`

ż t

0

eApt´sqBpXsq dWs. (7.48)

In addition, note that Theorem 4.8.5 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηqit holds that

ż t

0

›

›eApt´sqF pXsq›

›

LppP;¨Hr qds

ď F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

„ż t

0

pt´ sqpγ´η´rq ds

“ F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

tp1`γ´η´rq

p1` γ ´ η ´ rqă 8.

(7.49)

Moreover, observe that Theorem 4.8.5 ensures that for all t P r0, T s, r P rγ, β ` 12q

it holds that

„ż t

0

›

›eApt´sqBpXsq›

›

2

LppP;¨HSpU,Hrqqds

12

ď BLippHγ ,HSpU,Hβqqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

„ż t

0

pt´ sqp2β´2rq ds

12

“ BLippHγ ,HSpU,Hβqqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

tp12`β´rq

p1` 2β ´ 2rq12ă 8

(7.50)

Combining (7.48), (7.49), and (7.50) with Theorem 4.8.5 completes the proof ofProposition 7.1.16.


7.1.7 Temporal-regularity of solution processes of SPDEs

Exercise 7.1.17 (Temporal regularity*). Assume the setting in Subsection 7.1.2and let X : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s/BpHγq-predictable stochastic processwhich satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a pΩ,F ,P, pFtqtPr0,T sq-mildsolution of the SPDE


Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`

0,mint1`γ´η´r, 12`β´ru˘

it holds that

supt1,t2Pr0,T st1‰t2

¨

˝

›

›

`

Xt1 ´ et1AX0

˘

´`

Xt2 ´ et2AX0

˘›

›

LppP;¨Hr q

|t1 ´ t2|ε

˛

‚ă 8. (7.52)

Exercise 7.1.18 (Temporal regularity revisited*). Assume the setting in Subsec-tion 7.1.2, let δ P rγ,8q, assume that ξpΩq Ď Hδ and E

“

ξpHδ‰

ă 8, and letX : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s/BpHγq-predictable stochastic process which sat-isfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a pΩ,F ,P, pFtqtPr0,T sq-mild solutionof the SPDE


Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`

0,mint1`γ´η´r, 12`β´ru˘

it holds that

supt1,t2Pr0,T st1‰t2

¨

˝

|mintt1, t2u|maxtr`ε´δ,0u

Xt1 ´Xt2LppP;¨Hr q

|t1 ´ t2|ε

˛

‚ă 8. (7.54)

7.1.8 Existence of continuous solutions

See, e.g., Theorem 7.1 in Van Neerven et al. [22] for a more general result.


Proposition 7.1.19. Let pH, x¨, ÿH , ¨Hq and pU, x¨, ÿU , Üq be separable R-Hilbertspaces, let H Ď H be a non-empty orthonormal basis of H, let T P p0,8q, ρ P R,p P r2,8q, let pΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, letpWtqtPr0,T s be an IdU -cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, let A : DpAq ĎH Ñ H be a diagonal linear operator which satisfies sup

`

t´1u Y σP pA ´ ρq˘

ă

0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated toρ ´ A, and let γ P R, η P r0, 1q, F P LippHγ, Hγ´ηq, B P LippHγ, HSpU,Hγ´η2qq,ξ PMpF0,BpHγqq. Then there exists an pFtqtPr0,T s/BpHγq-adapted stochastic processX : r0, T sˆΩ Ñ Hγ with continuous sample paths which satisfies that for all t P r0, T s

it holds P-a.s. that Xt “ etAξ `şt

0ept´sqAF pXsq ds `

şt

0ept´sqABpXsq dWs and which

satisfies

suptPr0,T s

›

›max

1, XtHγ

(›

›

LppP;|¨|Rqď?

2›

›max

1, ξHγ(›

›

LppP;|¨|Rq

¨ Ep1´ηq„

T 1´η?


à


.(7.55)

Proof of Proposition 7.1.19. Throughout this proof let Ωn P F0, n P N0, be thesets with the property that for all n P N0 it holds that Ωn “ tξHγ ă nu and letξn : Ω Ñ Hγ, n P N, be the mappings with the property that for all n P N it holdsthat ξn “ ξ1Ωn . Note that for all q P r0,8q, n P N it holds that E

“

ξnqHγ

‰

ď nq ă 8.Theorem 7.1.11, Exercise 7.1.17, and the Kolmogorov-Chentsov theorem hence en-sure that there exist pFtqtPr0,T s/BpHγq-adapted stochastic processes with continuoussample paths Xn : r0, T s ˆ Ω Ñ Hγ, n P N, which satisfy suptPr0,T sE

“

Xnt Hγ

‰

ă 8

and which satisfy that for all n P N, t P r0, T s it holds P-a.s. that

Xnt “ etAξn `

ż t

0

ept´sqAF pXns q ds`

ż t

0

ept´sqABpXns q dWs. (7.56)

Observe that for all k P N, n,m P tk, k ` 1, . . . u, t P r0, T s it holds P-a.s. that

1Ωk rXnt ´X

mt s “

ż t

0

ept´sqA1Ωk

“

F`

1ΩkXns

˘

´ F`

1ΩkXms

˘‰

ds

`

ż t

0

ept´sqA1Ωk

“

B`

1ΩkXns

˘

´B`

1ΩkXms

˘‰

dWs.

(7.57)

We can hence apply Proposition 2.1 in [11] to obtain that for all k P N, n,m P

tk, k ` 1, . . . u it holds that

suptPr0,T s

1Ωk rXnt ´X

mt sLppP;Hγq

“ 0. (7.58)


This implies that

P

˜

@ k P N : @n,m P tk, k ` 1, . . . u : 1Ωk

«

suptPr0,T s

Xnt ´X

mt Hγ

ff

“ 0

¸

“ 1. (7.59)

Next let Y : r0, T s ˆ Ω Ñ Hγ be the mapping with the property that for all pt, ωq Pr0, T s ˆ Ω it holds that

Ytpωq “8ÿ

n“1

Xnt pωq ¨ 1ΩnzΩn´1pωq. (7.60)

Note that for all n P N it holds that

1Ωn suptPr0,T s

Yt ´Xnt Hγ “ sup

tPr0,T s

1ΩnYt ´ 1ΩnXnt Hγ

“ suptPr0,T s

›

›

›

›

›

«

nÿ

k“1

1ΩkzΩk´1Xkt

ff

´ 1ΩnXnt

›

›

›

›

›

Hγ

“ suptPr0,T s

›

›

›

›

›

nÿ

k“1

1ΩkzΩk´1

“

Xkt ´X

nt

‰

›

›

›

›

›

Hγ

“

nÿ

k“1

1ΩkzΩk´1

«

1Ωk suptPr0,T s

›

›Xkt ´X

nt

›

›

Hγ

ff

(7.61)

This and (7.59) show that

P

˜

@n P N : 1Ωn suptPr0,T s

Yt ´Xnt Hγ “ 0

¸

“ 1. (7.62)

Hence, we obtain that for all n P N, t P r0, T s it holds P-a.s. that

1ΩnYt “ 1ΩnXnt

“ 1Ωn

„

etAξn `

ż t

0

ept´sqAF pXns q ds`

ż t

0

ept´sqABpXns q dWs

“ 1Ωn

„

etAξ `

ż t

0

ept´sqA1ΩnF pXns q ds`

ż t

0

ept´sqA1ΩnBpXns q dWs

“ 1Ωn

„

etAξ `

ż t

0

ept´sqAF pYsq ds`

ż t

0

ept´sqABpYsq dWs

.

(7.63)

This implies that for all t P r0, T s it holds P-a.s. that

Yt “ etAξ `

ż t

0

ept´sqAF pYsq ds`

ż t

0

ept´sqABpYsq dWs. (7.64)

7.2. EXAMPLES OF SPDES 245

Next note that (7.62) and Proposition 7.1.15 ensure that for all n P N it holds that

suptPr0,T s

›

›max

1, 1ΩnYtHγ(›

›

LppP;|¨|q“ sup

tPr0,T s

›

›max

1, 1ΩnXnt Hγ

(›

›

LppP;|¨|q

ď suptPr0,T s

›

›max

1, Xnt Hγ

(›

›

LppP;|¨|qď?

2›

›max

1, ξnHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


à


.

(7.65)

This and Fatou’s lemma imply that for all t P r0, T s it holds that

›

›max

1, YtHγ(›

›

LppP;|¨|q“

›

›

›lim infnÑ8

max

1, 1ΩnYtHγ(

›

›

›

LppP;|¨|q

ď lim infnÑ8

›

›max

1, 1ΩnYtHγ(›

›

LppP;|¨|qď?

2›

›max

1, ξHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


à


.

(7.66)


7.2 Examples of SPDEs

7.2.1 Second order SPDEs*

Let T, ϑ P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pU, 〈¨, ¨〉U , Üq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, ξ P

L2pP|F0 ; ¨Hq, let pWtqtPr0,T s be an IdU -cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener pro-cess, let pekqkPN Ď H satisfy for all k P N that

ek ““

p?

2 sinpkπxqqxPp0,1q‰

Borelp0,1q,BpRq, (7.67)

let A : DpAq Ď H Ñ H be the linear operator which satisfies for all v P DpAq that

DpAq “

#

w P H :8ÿ

k“1

k4|〈ek, w〉H |

2Ră 8

+

(7.68)

and

Av “8ÿ

k“1

´ϑπ2k2 〈ek, v〉H ek, (7.69)


let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to Á,let f, b : p0, 1q ˆ RÑ R be Bpp0, 1q ˆRq/BpRq-measurable functions with

ż 1

0

|fpx, 0q|2 ` |bpx, 0q|2 dx` supxPp0,1q

supy1,y2PRy1‰y2

|fpx,y1q´fpx,y2q|`|bpx,y1q´bpx,y2q||y1ý2|

ă 8, (7.70)

let β P p´12,´14q, and let F : H Ñ H be the function which satisfies for all v PL2pBorelp0,1q; |¨|Rq that

F prvsBorelp0,1q,BpRqq ““`

fpx, vpxqq˘

xPp0,1q

‰

Borelp0,1q,BpRq. (7.71)

Observe that the linear operator A is the Laplace operator with Dirichlet boundaryconditions on L2pBorelp0,1q; |¨|Rq multiplied by ϑ. Next observe that for all v P H itholds that

8ÿ

k“1

bp¨, vp¨qq ekp¨q2Hβ“

8ÿ

k“1

›

›pÁqβ`

bp¨, vp¨qq ekp¨q˘›

›

2

H

“

8ÿ

l,k“1

ˇ

ˇ

⟨el, pÁq

β`

bp¨, vp¨qq ekp¨q˘⟩

H

ˇ

ˇ

2

R“

8ÿ

l,k“1

ˇ

ˇ

⟨pÁqβel, bp¨, vp¨qq ekp¨q

⟩H

ˇ

ˇ

2

R

“

8ÿ

l,k“1

›

›pÁqβel›

›

2

H|〈el, bp¨, vp¨qq ekp¨q〉H |

2R

“ÿ

lPN

›

›pÁqβel›

›

2

H

«

ÿ

kPN

|〈ek, bp¨, vp¨qq elp¨q〉H |2

ff

“ÿ

lPN

›

›pÁqβel›

›

2

Hbp¨, vp¨qq elp¨q

2H ď 2 bp¨, vp¨qq2H

«

ÿ

lPN

›

›pÁqβel›

›

2

H

ff

“ 2›

›pÁqβ›

›

2

HSpHqbp¨, vp¨qq2H ă 8.

(7.72)

This and Proposition 2.4.22 ensure that there exists a unique mapping B : H Ñ

HSpH,Hβq which satisfies for all v P L2pBorelp0,1q; |¨|Rq and all uniformly continousfunctions u : p0, 1q Ñ R that

B`

rvsBorelp0,1q,BpRq˘

rusBorelp0,1q,BpRq ““

pbpx, vpxqq ¨ upxqqxPp0,1q‰

Borelp0,1q,BpRq. (7.73)

In addition, (7.72) implies that for all v P H it holds that

BpvqHSpU,Hβq ď?

2›

›pÁqβ›

›

HSpHqbp¨, vp¨qqH ă 8. (7.74)


Moreover, note that for all v, w P H it holds that

Bpvq ´Bpwq2HSpU,Hβq “ÿ

kPN

rBpvq ´Bpwqs ek2Hβ

“ÿ

kPN

›

›pÁqβ rrBpvq ´Bpwqs eks›

›

2

H

“ÿ

k,lPN

ˇ

ˇ

⟨el, pÁq

βrrBpvq ´Bpwqs eks

⟩H

ˇ

ˇ

2

“ÿ

k,lPN

›

›pÁqβel›

›

2

H|〈el, rBpvq ´Bpwqs ek〉H |

2 .

(7.75)

This proves that for all v, w P H it holds that

Bpvq ´Bpwq2HSpU,Hβq “ÿ

k,lPN

›

›pÁqβel›

›

2

H|〈ek, rBpvq ´Bpwqs el〉H |

2

“ÿ

lPN

›

›pÁqβel›

›

2

HrBpvq ´Bpwqs el

2H

ď 2ÿ

lPN

›

›pÁqβel›

›

2

Hbp¨, vp¨qq ´ bp¨, wp¨qq2H

ď 2›

›pÁqβ›

›

2

HSpHq

»

– supxPp0,1q

supy1,y2PRy1‰y2

|bpx, y1q ´ bpx, y2q|

|y1 ´ y2|

fi

fl

2

v ´ w2H ă 8.

(7.76)

This shows that the mapping B : H Ñ HSpH,Hβq in (7.73) is an element of theset C0,1pH,HSpH,Hβqq. We can hence apply Theorem 7.1.11 to obtain that thereexists an up to modifications unique pFtqtPr0,T s/BpHq-predictable stochastic pro-cess X : r0, T s ˆ Ω Ñ H which satisfies suptPr0,T sE

“

Xt2H

‰

ă 8 and which is apΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


The stochastic process X is thus a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXtpxq “”

ϑ B2

Bx2Xtpxq ` fpx,Xtpxqq

ı

dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0

(7.78)with X0pxq “ ξpxq for x P p0, 1q, t P r0, T s. For example, in the case where b fulfills@x P p0, 1q, y P R : bpx, yq “ 1, the SPDE (7.78) reduces to the SPDE with additivenoise

dXtpxq “”

ϑ B2

Bx2Xtpxq ` fpx,Xtpxqq

ı

dt` dWtpxq, Xtp0q “ Xtp1q “ 0 (7.79)


with X0pxq “ ξpxq for x P p0, 1q, t P r0, T s and in the case where f and b fulfill@x P p0, 1q, y P R : fpx, yq “ 0 and bpx, yq “ y, the SPDE (7.78) reduces to thestochastic heat equation with linear multiplicative noise

dXtpxq “”

ϑ B2

Bx2Xtpxq

ı

dt`Xtpxq dWtpxq (7.80)

with X0pxq “ ξpxq for x P p0, 1q, t P r0, T s. In the literature the SPDE (7.80) isreferred to as continuous version of the parabolic Anderson model.

Lemma 7.2.1. Let α, β P R and let φ, ψ : p1,8q Ñ R be the functions which satisfyfor all x P p1,8q that φpxq “ α rlnpxqsβ and ψpxq “ lnplnpxqq. Then it holds for allx P p1,8q that φ, ψ P C8pR,Rq and

φ1pxq “αβ

x rlnpxqsp1´βqand ψ1pxq “

1

x lnpxq. (7.81)


Class exercise 7.2.2 (On optimal regularity statements*). Let T P p0,8q, letpH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let A :

DpAq Ď H Ñ H be the Laplace operator with Dirichlet boundary conditions onH, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated toÁ, and let Hr Ď YsPRHs, r P R, be the sets which satisfy for all r P R thatHr “ YsPpr,8qHs. Prove or disprove the following the statement: There exists a realnumber r P R such that Hr “ Hr.

Class exercise 7.2.3 (*). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq,

let ξ P H, let pWtqtPr0,T s be an IdH-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, letb : p0, 1qˆRÑ R be a globally Lipschitz continuous function, let X : r0, T s ˆΩ Ñ Hbe a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq

for x P p0, 1q, t P r0, T s, let A : DpAq Ď H Ñ H be the Laplace operator withDirichlet boundary conditions on H, and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to Á.

(i) For which r P R does it holds that @ t P r0, T s : PpXt P Hrq “ 1?

(ii) For which r P R does it holds that @ t P p0, T s : PpXt P Hrq “ 1?


Exercise 7.2.4 (*). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis,let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let

pWtqtPr0,T s be an IdH-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, and let X : r0, T sˆΩ Ñ H be a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ 0 (7.82)

for x P p0, 1q, t P r0, T s. Prove or disprove the following statement: It holds thatş1

0E“

|XT pxq|2‰

dx “ř8

n“11é´2π2n2T

π2n2 .


Part IV

Numerical Analysis of SPDEs

251

Chapter 8

Strong numerical approximationsfor SPDEs

Consider the setting of Section 7.1.2. If one wants to simulate a solution process ofan SPDE on a computer approximatively, then one needs to discretize the possiblyinfinite dimensional Hilbert space H (spatial approximations), the possibly infinitedimensional Hilbert space U (noise approximations) as well as the time intervalr0, T s (temporal approximations). Section 8.1 deals with spatial approximations forSPDEs, Section 8.2 analyses temporal numerical approximations for SPDEs, Sec-tion 8.3 considers noise approximations for SPDEs, and Section 8.4 combines spatial(Section 8.1), temporal (Section 8.2), and noise approximations (Section 8.3) to ob-tain full-discrete numerical approximations for SPDEs.

8.1 Spatial spectral Galerkin approximations for

SPDEs

8.1.1 Galerkin projections

We study Galerkin approximations in Hilbert spaces. For this we recall the conceptof a projection of a Hilbert space on a subspace in Definition 3.4.12 above.

253

254 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES

Lemma 8.1.1 (Representations of projections in Hilbert spaces*). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let U Ď H be a closed subspace of H, letB Ď U be an orthonormal basis of U , and let v P H. Then

PU,Hpvq “ P spanpBq,H pvq “ÿ

bPB

〈b, v〉H b. (8.1)

8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 255

Example 8.1.2 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letB Ď H be an orthonormal basis of H, let In Ď B, n P N, be a non-decreasingsequence of subsets of B which satisfies YnPNIn “ B, and let v P H. Then it holdsfor all n P N that

›

›

›

›

›

v ´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“

›

›

›

›

›

«

ÿ

bPB

〈b, v〉H b

ff

´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“

›

›

›

›

›

›

ÿ

bPBzIn

〈b, v〉H b

›

›

›

›

›

›

H

“

d

ÿ

bPBzIn

〈b, v〉H b2H “

d

ÿ

bPBzIn

|〈b, v〉H |2.

(8.2)

The fact thatř

bPB |〈b, v〉H |2“ v2H ă 8 and the assumption that In Ď B, n P N, is

non-decreasing with YnPNIn “ B hence imply that

lim supnÑ8

›

›

›

›

›

v ´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“ 0. (8.3)

It thus holds for all sufficiently large n P N thatÿ

bPIn

〈b, v〉H b (8.4)

is, in the sense of (8.3), a good approximation of v P H. For example, assume thatK “ R, assume that

pH, 〈¨, ¨〉H , ¨Hq “`

L2pBorelp0,1q; |¨|Rq, ¨L2pBorelp0,1q;|¨|Rq

, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

˘

, (8.5)

let en P H, n P N, satisfy for all n P N that en ““

p?


Borelp0,1q,BpRq,

assume that B “ te1, e2, . . . u, assume for all n P N that In “ te1, e2, . . . , enu, and letw P L2pBorelp0,1q; |¨|Rq. Then

lim supnÑ8

ż 1

0

ˇ

ˇ

ˇ

ˇ

wpxq ńř

k“1

2 sinpkπxq

„

1

∫0

sinpkπyqwpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx “ 0 (8.6)

and it holds for all sufficiently large n P N that the function

nř

k“1

2 sinpkπxq

„

1

∫0

sinpkπyqwpyq dy

, x P p0, 1q, (8.7)

is, in the sense of (8.6), a good approximation of w : p0, 1q Ñ R.


Example 8.1.2 above illustrates how suitable finite-dimensional approximationsof vectors in an infinite dimensional Hilbert space can be obtained; see (8.4) in Ex-ample 8.1.2. Equation (8.3) in Example 8.1.2 also shows that these approximationsin finite dimensional subspaces of the Hilbert space converge to the original vectorin the infinite dimensional Hilbert space. In Proposition 8.1.4 below we intend toprovide more information about how fast the finite dimensional approximations con-verge to the vector in the infinite dimensional Hilbert space. In the formulation ofProposition 8.1.4 the following lemma is used.

Lemma 8.1.3 (*). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with infpσP pAqq ą 0,let B Ď H be an orthonormal basis of H, let λ : B Ñ p0,8q be a function whichsatisfies for all v P DpAq that

DpAq “

#

w P H :ÿ

bPB

|λb 〈b, w〉H |2ă 8

+

(8.8)

and Av “ř

bPB λb 〈b, v〉H b, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolationspaces associated to A, and let r P R, v P Hr, b P B. Then

〈b, v〉H b “@

bpλbqr

, vD

Hr

bpλbqr

P Hr. (8.9)

The proof of Lemma 8.1.3 is clear and therefore omitted. Instead we now formu-late the main result of this subsection.


Proposition 8.1.4 (A central idea for spectral Galerkin approximations*). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0, let B Ď H be an or-thonormal basis of H, let λ : BÑ p0,8q be a function which satisfies for all v P DpAqthat

DpAq “

#

w P H :ÿ

bPB

|λb 〈b, w〉H |2ă 8

+

(8.10)

and Av “ř

bPB λb 〈b, v〉H b, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolationspaces associated to A, and let r P R, ρ P r0,8q, I P PpBq, πI P LpHrq satisfy forall v P Hr that

πIpvq “ÿ

bPI

〈b, v〉H b “ÿ

bPI

@

bpλbqr

, vD

Hr

bpλbqr

“ PspanpIq

Hr,Hrpvq P Hr. (8.11)

Then it holds for all v P Hr`ρ that

v ´ πIpvqHr ď›

›A´ρ pIdHr ´πIq›

›

LpHrqvHr`ρ “

„

infbPBzI

λb

´ρ

vHr`ρ . (8.12)

Proof of Proposition 8.1.4*. Observe that Proposition 3.6.6 implies that

›

›A´ρ pIdHr ´πIq›

›

LpHrq“

›

›

›

›

›

›

A´ρ

¨

˝

ÿ

bPBzI

@

bpλbqr

, p¨qD

Hr

bpλbqr

˛

‚

›

›

›

›

›

›

LpHrq

“

›

›

›

›

›

›

ÿ

bPBzI

1pλbqρ

@

bpλbqr

, p¨qD

Hr

bpλbqr

›

›

›

›

›

›

LpHrq

“ supbPBzI

”

1pλbqρ

ı

“

„

infbPBzI

λb

´ρ

.

(8.13)


In a number of cases the right hand side of estimate (8.12) converges to zero witha polynomial rate of convergence. This is illustrated in the next example.


Example 8.1.5 (The Laplace operator with Dirichlet boundary conditions*). LetA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator withDirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R,be a family of interpolation spaces associated to Á, let r P R, ρ P r0,8q, andlet en P L

2pBorelp0,1q; |¨|Rq, n P N, be the vectors which satisfy for all n P N that

en “ rp?

2 sinpnπxqqxPp0,1qsBorelp0,1q,BpRq. Then Proposition 8.1.4 proves that for allv P Hr`ρ, n P N it holds that

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

ďvHr`ρπ2ρ n2ρ

ďvHr`ρn2ρ

. (8.14)

Note that, in the setting of Example 8.1.5, it holds for all v P Hr`ρ that

supnPN

˜

n2ρ

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

ă 8. (8.15)

The polynomial convergence rate 2ρ in (8.14) and (8.15) can, in general, not beimproved. This is the subject of the next exercise.

Exercise 8.1.6 (Lower bounds on the convergence speed of spectral Galerkin projec-tions*). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace oper-ator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be a family of interpolation spaces associated to Á, let r P R, ρ P r0,8q,and let en P L

2pBorelp0,1q; |¨|Rq, n P N, be the vectors which satisfy for all n P N that

en “ rp?

2 sinpnπxqqxPp0,1qsBorelp0,1q,BpRq. Give an example of a vector v P Hr`ρ suchthat for all ε P p0,8q it holds that

supnPN

˜

np2ρ`εq

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

“ 8. (8.16)

The next result, Corollary 8.1.7, specialises Proposition 8.1.4 to the case wherethe vector in the possibly infinite dimensional Hilbert space is the solution processof some SPDE at a fixed time instance.


Corollary 8.1.7 (Galerkin projections for SPDEs*). Assume the setting in Sub-section 7.1.2, let B Ď H be an orthonormal basis of H, let λ : B Ñ p´8, 0q be afunction which satisfies for all v P DpAq that

DpAq “

#

w P H :ÿ

bPB

|λb 〈b, w〉H |2ă 8

+

(8.17)

and Av “ř

bPB λb 〈b, v〉H b, let r P p´8, γs, I P PpBq, πI P LpHrq satisfy for allv P Hr that

πIpvq “ÿ

bPI

〈b, v〉H b “ÿ

bPI

@

b|λb|

r , vD

Hr

b|λb|

r “ PspanpIq

Hr pvq P Hr, (8.18)

and let X : r0, T s ˆ Ω Ñ Hγ be an up to modifications unique pFtqtPr0,T s/BpHγq-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and whichis a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


Then

suptPr0,T s

Xt ´ πIpXtqLppP;¨Hr qď

„

infbPBzI

|λb|

pr´γq

suptPr0,T s

XtLppP;¨Hγ qă 8. (8.20)

Corollary 8.1.7 is an immediate consequence from Proposition 8.1.4 and Propo-sition 7.1.13.

Class exercise 8.1.8 (*). Let f : p0, 1q Ñ R be the function which satisfies for allx P p0, 1q that fpxq “ x. For which r P R does it hold that

supNPN

˜

N r

ż 1

0

ˇ

ˇ

ˇ

ˇ

fpxq Ńř

n“1

2 sinpnπxq1ş

0

sinpnπyq fpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx

¸

ă 8? (8.21)

Class exercise 8.1.9 (*). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rqbe the Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq. Forwhich r P R does it hold that

supNPN

supvPDpAq,AvHď1

supϕPv

˜

N r

ż 1

0

ˇ

ˇ

ˇ

ˇ

ϕpxq Ńř

n“1

2 sinpnπxq1ş

0

sinpnπyqϕpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx

¸

ă 8?

(8.22)


8.1.2 Setting

Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letλ PMpB,Rq satisfy for all v P DpAq that

DpAq “

#

w P H :ÿ

bPB

|λb 〈eb, w〉H |2ă 8

+

(8.23)

and Av “ř

bPB λb 〈b, v〉H b, and let pπIqIPPpBq Ď LpHγ´ηq satisfy for all v P Hγ´η,I P PpBq that

πIpvq “ÿ

bPI

〈b, v〉H b. (8.24)

The above setting allows us to apply Theorem 7.1.11 to obtain that for every I PPpBq there exist an up to modifications unique pFtqtPr0,T s/BpπIpHγqq-predictablestochastic processes XI : r0, T s ˆ Ω Ñ πIpHγq which satisfies for all t P r0, T s thatsupsPr0,T s X

Is LppP;¨Hγ q

ă 8 and

“

XIt ´ e

AtπIpξq‰

P,BpπIpHγqq“

ż t

0

eApt´sqπI`

F pXIs q˘

ds`

ż t

0

eApt´sqπI`

BpXIs q dWs

˘

.

(8.25)

8.1.3 A strong numerical approximation result for spectralGalerkin approximations of SPDEs

Lemma 8.1.10 (*). Assume the setting in Section 8.1.2 and let I, J P PpBq. Then

suptPr0,T s

›

›XIt ´X

Jt

›

›

LppP;¨Hγ qď?

2 suptPr0,T s

›

›πIzJXIt ` πJzIX

Jt

›

›

LppP;¨Hγ q(8.26)

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η

à

ppp´ 1qT 1´η |πIXJB|C0,1pHγ ,HSpU,Hγ´η2qq

ă 8.

Proof of Lemma 8.1.10*. Let F P CpHγ, Hγ´ηq and B P CpHγ, HSpU,Hγ´η2qq bethe functions which satisfy for all v P Hγ, u P U that

F pvq “ πIXJ`

F pvq˘

and Bpvqu “ πIXJ`

Bpvqu˘

. (8.27)


Next note that the fact that πIπJ “ πIXJ shows that for all t P r0, T s it holds that

„

“

XIt

‰

P,BpHγq´

ż t

0

eApt´sqF pXIs qds´

ż t

0

eApt´sqBpXIs q dWs

`

„ż t

0

eApt´sqF pXJs qds`

ż t

0

eApt´sqBpXJs q dWs ´

“

XJt

‰

P,BpHγq

“

„

“

XIt

‰

P,BpHγq

´ πJ

ˆ

“

eAtπIpξq‰

P,BpHγq`

ż t

0

eApt´sqπIF pXIs qds`

ż t

0

eApt´sqπIBpXIs q dWs

˙

`

„

πI

ˆ

“

eAtπJpξq‰

P,BpHγq`

ż t

0

eApt´sqπJF pXJs qds`

ż t

0

eApt´sqπJBpXJs q dWs

˙

´“

XJt

‰

P,BpHγq

““

IdHγ´η ´πJ‰ “

XIt

‰

P,BpHγq`“

πI ´ IdHγ´η‰ “

XJt

‰

P,BpHγq

““

πIzJpXIt q ´ πJzIpX

Jt q‰

P,BpHγq.

(8.28)

An application of Proposition 7.1.6 hence proves that

suptPr0,T s

›

›XIt ´X

Jt

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à

ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq

¨?

2 suptPr0,T s

›

›πIzJpXIt q ´ πJzIpX

Jt q›

›

LppP;¨Hγ q.

(8.29)

This and the identity

suptPr0,T s

›

›πIzJpXIt q ´ πJzIpX

Jt q›

›

LppP;¨Hγ q“ sup

tPr0,T s

›

›πIzJpXIt q ` πJzIpX

Jt q›

›

LppP;¨Hγ q

(8.30)complete the proof of Lemma 8.1.10.

The next result, Corollary 8.1.11, is an immediate consequence of Lemma 8.1.10above.


Corollary 8.1.11 (*). Assume the setting in Section 8.1.2, let r P pγ,mintγ ` 1 ´η, β ` 12uq, I P PpBq, and assume that ξpΩq Ď Hr and E

“

ξpHr‰

ă 8. Then

suptPr0,T s

›

›XB

t ´XIt

›

›

LppP;¨Hγ qď?

2“

infbPBzI |λb|‰pγ´rq

suptPr0,T s

›

›XB

t

›

›

LppP;¨Hr q(8.31)

¨ Ep1´ηq„

T 1´η?


à

ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq

ă 8.

The next result, Corollary 8.1.12, establishes a certain uniform moment boundfor the processes XI , I P PpBq. Corollary 8.1.12 is an immediate consequence ofProposition 7.1.13 and of Theorem 7.1.11.

Corollary 8.1.12 (*). Assume the setting in Section 8.1.2, let r P rγ,mintγ ` 1 ´η, β ` 12uq, ρ P

“

maxtr ` η ´ γ, 2pr ´ βqu, 1˘

, and assume that ξpΩq Ď Hr andE“

ξpHr‰

ă 8. Then

supIPPpBq

suptPr0,T s

›

›XIt

›

›

LppP;¨Hr qď?

2 max

1, ξLppP;¨Hr q

(

¨ Ep1´ρq„

T 1´ρ?

2 F LippHr,Hr´ρq?1´ρ

à

T 1´ρppp´ 1q BLippHr,HSpU,Hr´ρ2qq

ă 8.(8.32)

Lemma 8.1.13. Assume the setting in Section 8.1.2, let r P rγ,mintγ`1´η, β`12uq,I, J P PpBq, and assume that ξpΩq Ď Hr and E

“

ξpHr‰

ă 8. Then

suptPr0,T s

›

›XIt ´X

Jt

›

›

LppP;¨Hγ qď 2

›

›pÁqpγ´rqπI4J›

›

LpHqmaxKPtI,Ju

suptPr0,T s

›

›XKt

›

›

LppP;¨Hr q

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η

à


ă 8.


Proof of Lemma 8.1.13. Observe that

suptPr0,T s

›


Jt

›

›

2

LppP;¨Hγ qq“ sup

tPr0,T s

›


Jt

›

›

2

LppP;¨Hγ q

“ suptPr0,T s

›

›pÁqpγ´rqπI4J pÁqpr´γq

“

πIzJXIt ` πJzIX

Jt

‰›

›

2

LppP;¨Hγ q

ď›


›

2

LpHqsuptPr0,T s

›


Jt

›

›

2

LppP;¨Hr q

“›


›

2

LpHqsuptPr0,T s

›

›

›

›


Jt

›

›

2

Hr

›

›

›

Lp2pP;|¨|Rq

“›


›

2

LpHqsuptPr0,T s

›

›

›

›

›πIzJXIt

›

›

2

Hr`›

›πJzIXJt

›

›

2

Hr

›

›

›

Lp2pP;|¨|Rq

ď›


›

2

LpHqsuptPr0,T s

„

›

›

›

›

›πIzJXIt

›

›

2

Hr

›

›

›

Lp2pP;|¨|Rq`

›

›

›

›

›πJzIXJt

›

›

2

Hr

›

›

›

Lp2pP;|¨|Rq

“›


›

2

LpHqsuptPr0,T s

”

›

›πIzJXIt

›

›

2

LppP;¨Hr q`›

›πJzIXJt

›

›

2

LppP;¨Hr q

ı

.

(8.33)

Combining this with Lemma 8.1.10 completes the proof of Lemma 8.1.13.

Let us illustrate a bit different perspective on Lemma 8.1.13. For this assume thesetting in Section 8.1.2, let dr : PpBq ˆ PpBq Ñ r0,8q, r P p0,8q, be the mappingswhich satisfy for all I, J P PpBq, r P p0,8q that

drpI, Jq “›

›pÁq´rπI4J›

›

LpHq, (8.34)

let r P pγ,mint1` γ ´ η, β ` 12uq, and assume ξpΩq Ď Hr and E“

ξpHr‰

ă 8. Exer-cise 8.1.14 then shows that the pair pPpBq, drq is a metric space and Lemma 8.1.13,in particular, ensures that the mapping1

pPpBq, dr´γq Q I ÞÑ XIP

´

Cpr0, T s, LppP; ¨Hγ qq, ¨Cpr0,T s,LppP;¨Hγ qq

¯

(8.35)

is globally Lipschitz continuous with a Lipschitz constant which is smaller or equal

1Clearly, the domain (the set of arguments) of the mapping (8.35) is not pPpBq, dr´γq but theset PpBq. The notation (8.35) is nonetheless used in order to emphasize the specific metric definedon the set PpBq of arguments. The same comment applies to the co-domain of the mapping (8.35).


than

2 ¨

«

supIPPpBq

suptPr0,T s

›

›XIt

›

›

LppP;¨Hr q

ff

(8.36)

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |CC1pHγ,Hγ´ηq?1´η

à


ă 8.

Exercise 8.1.14 (*). Assume the setting in Section 8.1.2 and let dr : PpBqˆPpBq Ñr0,8q, r P p0,8q, be the mappings which satisfy for all I, J P PpBq, r P p0,8q that

drpI, Jq “›

›pÁq´rπI4J›

›

LpHq. (8.37)

Prove that for every r P p0,8q it holds that the pair pPpBq, drq is a metric space.

Exercise 8.1.15 (Numerical estimates for strong convergence rates*). Let T “ 1,let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let

A : DpAq Ď H Ñ H be the Laplace operator with Dirichlet boundary conditions onH (see Definition 3.6.10), let en P H, n P N, satisfy for all n P N that en “rp?

2 sinpnπxqqxPp0,1qsP,BpHq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s

be an IdH-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, let X : r0, T s ˆ Ω Ñ H bean pFtqtPr0,T s/BpHq-predictable stochastic process which fulfills for all t P r0, T s that

rXtsP,BpHq “

ż t

0

eApt´sq dWs, (8.38)

let πN P LpHq, N P N, satisfy for all N P N, v P H that πNpvq “řNn“1 〈en, v〉H en,

and let XN : r0, T s ˆ Ω Ñ πNpHq, N P N, be pFtqtPr0,T s/BpπNpHqq-predictablestochastic processes which satisfy for all N P N, t P r0, T s that

“

XNt

‰

P,BpHq “

ż t

0

πN eApt´sq dWs. (8.39)

Write a Matlab function which plots Monte Carlo approximations of the real num-

bers`

E“

X223

T ´ XNT

2H

‰˘12for N P t20, 21, 22, 23, . . . , 218, 219u. Hint: Use the fact

that for every N P N you can simulate exactly from XNT pPqBpHq.

Class exercise 8.1.16 (Convergence speed of spectral Galerkin approximations*).Assume the setting in Section 8.1.2 and assume that ξpΩq Ď Hγ`1 and E

“

ξpHγ`1

‰

ă

8. For which r P R does it holds that there exists a real number C P R such that forall I P PpBqztBu it holds that

suptPr0,T s

›

›XB

t ´XIt

›

›

LppP;¨Hγ qď C

“

infbPBzI |λb|‰´r

. (8.40)

8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 265

8.2 Temporal numerical approximations for SPDEs

Using Definition 6.3.31 and Definition 6.3.32 in Subsection 6.3.5 we now present inSubsections 8.2.1 and 8.2.3 below a few temporal numerical approximation meth-ods for SPDEs and then analyze the strong approximation errors for one of theseapproximation methods in Subsection 8.2.4 below.

8.2.1 Euler type approximations for SPDEs

8.2.1.1 Exponential Euler method

Definition 8.2.1 (Exponential Euler approximations*). Assume the setting in Sec-tion 7.1.2 and let N P N. Then we say that Y is a pΩ,F ,P, pFtqtPr0,T sq-exponentialEuler approximation for the SPDE


with time step size TN if and only if it holds that Y P Mpt0, 1, . . . , Nu ˆ Ω, Hγq

is a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic process which fulfills for all n P

t0, 1, . . . , N ´ 1u that Y0 “ ξ and

rYn`1sP,BpHγq “”

eATN

`

Yn ` F pYnqTN

˘

ı

P,BpHγq`

ż pn`1qTN

nTN

eATNBpYnq dWs. (8.42)

Definition 8.2.2 (Naturally-interpolated exponential Euler approximations*). As-sume the setting in Section 7.1.2 and let h P p0,8q. Then we say that Y is apΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation for theSPDE


with time step size h if and only if it holds that Y P Mpr0, T s ˆ Ω, Hγq is anpFtqtPr0,T s/BpHγq-adapted stochastic process which fulfills for all t P p0, T s that Y0 “ ξand

rYtsP,BpHγq

““

eApt´zthq`

Yzth ` F pYzthq pt´ zthq˘‰

P,BpHγq`

ż t

zth

eApt´zthqB`

Yzth

˘

dWs.(8.44)


Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ

be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation forthe SPDE


with time step size h. Then note that for all t P r0, T s it holds that

rYtsP,BpHγq “

„

eAtξ `

ż t

0

eApt´tsuhqF pYtsuhq ds

P,BpHγq`

ż t

0

eApt´tsuhqBpYtsuhq dWs.

(8.46)

Exercise 8.2.3 (*). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq bethe Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, letT P p0,8q, N P N, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s bean IdH-cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, let Y : r0, T s ˆ Ω Ñ H bea pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation for theSPDE

dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (8.47)

with time step size TN, and let X : r0, T sˆΩ Ñ H be an pFtqtPr0,T s/BpHq-predictablestochastic process which fulfills for all t P r0, T s that

rXtsP,BpHq “

ż t

0

eApt´sq dWs. (8.48)

Prove that for all r P r0, 14q it holds that`

E“

XT ´ YT 2H

‰˘12ď T r

p1´4rqNr .


8.2.1.2 Accelerated exponential Euler method

Definition 8.2.4 (Accelerated exponential Euler approximations*). Assume the set-ting in Section 7.1.2 and let N P N. Then we say that Y is a pΩ,F ,P, pFtqtPr0,T sq-accelerated exponential Euler approximation for the SPDE



is a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic process which fulfills for all n P

t0, 1, . . . , N ´ 1u that Y0 “ ξ and

rYn`1sP,BpHγq

“

”

eATN Yn ` A

´1`

eATN ´ IdH

˘

F pYnqı

P,BpHγq`

ż pn`1qTN

nTN

eAppn`1qTN´sqBpYnq dWs.

(8.50)

Definition 8.2.5 (Naturally-interpolated accelerated exponential Euler approxima-tions*). Assume the setting in Section 7.1.2 and let h P p0,8q. Then we say that Yis a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated accelerated exponential Euler approx-imation for the SPDE


with time step size h if and only if it holds that Y P Mpr0, T s ˆ Ω, Hγq is apFtqtPr0,T s/BpHγq-adapted stochastic process which fulfills for all t P p0, T s that Y0 “ ξand

rYtsP,BpHγq

“

„

eApt´zthq Yzth `

ż t

zth

eApt´sqF`

Yzth

˘

ds

P,BpHγq`

ż t

zth

eApt´sqB`

Yzth

˘

dWs.

(8.52)

Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ bea pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated accelerated exponential Euler approxi-mation for the SPDE



rYtsP,BpHγq “

„

eAtξ `

ż t

0

eApt´sqF`

Ytsuh

˘

ds

P,BpHγq`

ż t

0

eApt´sqB`

Ytsuh

˘

dWs. (8.54)


8.2.1.3 Linear-implicit Euler method

Definition 8.2.6 (Linear-implicit Euler approximations*). Assume the setting inSection 7.1.2 and let N P N. Then we say that Y is a pΩ,F ,P, pFtqtPr0,T sq-linear-implicit Euler approximation for the SPDE



is a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic processes which fulfills for all n Pt0, 1, . . . , N ´ 1u that Y0 “ ξ and


`

IdH ´TNA˘´1 `

Yn ` F`

Yn˘

TN

˘

ı

P,BpHγq

`

ż pn`1qTN

nTN

`

IdH ´TNA˘´1

B`

Yn˘

dWs.(8.56)

Definition 8.2.7 (Naturally-interpolated linear-implicit Euler approximations*).Assume the setting in Section 7.1.2 and let h P p0,8q. Then we say that Y isa pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Euler approximation forthe SPDE


with time step size h if and only if it holds that Y P Mpr0, T s ˆ Ω, Hγq is apFtqtPr0,T s/BpHγq-adapted stochastic processes which fulfills for all t P p0, T s thatY0 “ ξ and

rYtsP,BpHγq ““

pIdH ´pt´ zthqAq´1

`

Yzth ` F`

Yzth

˘

pt´ zthq˘‰

P,BpHγq

`

ż t

zth

pIdH ´pt´ zthqAq´1B

`

Yzth

˘

dWs.(8.58)


be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Euler approximationfor the SPDE




Yt “`

IdH ´pt´ ttuhqA˘´1 `

IdH ´hA˘´ttuhh

ξ

`

ż t

0

`


IdH ´hA˘ptsuh´ttuhqh

F`

Ytsuh

˘

ds

`

ż t

0

`



B`

Ytsuh

˘

dWs.

(8.60)

8.2.1.4 Linear-implicit Crank-Nicolson-Euler method

Definition 8.2.8 (Linear-implicit Crank-Nicolson-Euler approximations). Assumethe setting in Section 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be apFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic process which fulfills Y0 “ ξ and whichfulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´T

2NA˘´1

˜

`

IdH `T

2NA˘

Yn ` F pYnqTN`

ż pn`1qTN

nTN

BpYnq dWs

¸

.

(8.61)

Then we call Y a linear-implicit Crank-Nicolson-Euler approximation for the SPDE


with time step size TN.

Definition 8.2.9 (Naturally-interpolated linear-implicit Crank-Nicolson-Euler ap-proximations). Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ be a pFtqtPr0,T s/BpHγq-adapted stochastic processes which fulfills Y0 “ ξ andwhich fulfills that for all t P p0, T s it holds P-a.s. that

Yt “´

IdH ´pt´zthq

2A¯´1

ˆ

´

IdH `pt´zthq

2A¯

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

.

(8.63)

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Crank-Nicolson-Euler approximation for the SPDE


with time step size h.



be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Euler approximationfor the SPDE


with time step size h. Then note that for all t P r0, T s it holds P-a.s. that

Yt “`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘´ttuhh

ξ

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘ptsuh´ttuhqh

“

12AYtsuh ` F

`

Ytsuh

˘‰

ds

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


B`

Ytsuh

˘

dWs.

(8.66)

8.2.2 Nonlinearity-stopped Euler type approximations forSPDEs

The next result is a special case of Theorem 2.1 in Hutzenthaler et al. [8] (see also[9]).

Theorem 8.2.10 (Strong and weak divergence of the Euler method for SDEs withsuperlinearly growing coefficients). Let T, ε, p P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let W : r0, T s ˆ Ω Ñ R be a standard Brownian motion w.r.t.pFtqtPr0,T s, let µ, σ PMpBpRq,BpRqq, ξ PMpF0,BpRqq satisfy P

`

σpξq ‰ 0˘

ą 0, letY N : t0, 1, . . . , Nu ˆΩ Ñ R, N P N, satisfy that for all N P N, n P t0, 1, . . . , N ´ 1uit holds that Y N

0 “ ξ and

Y Nn`1 “ Y N

n ` µpY Nn q

TN` σpY N

n q`

Wpn`1qTN ´WnTN

˘

, (8.67)

and assume that for all x P p´8, 1εsYr1ε,8q it holds that |µpxq|`|σpxq| ě ε |x|p1`εq.Then limNÑ8 E

“

|Y NN |

p‰

“ 8.


8.2.2.1 Nonlinearity-stopped exponential Euler method

Definition 8.2.11 (Nonlinearity-stopped exponential Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochasticprocess which fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u itholds P-a.s. that

Yn`1 “ 1tF pYnq2HαąNTuYn

` 1tF pYnq2HαďNTueA

TN

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

.(8.68)

Then we call Y a nonlinearity-stopped exponential Euler approximation for the SPDE



Definition 8.2.12 (Naturally-interpolated nonlinearity-stopped exponential Eulerapproximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ ´ η, γs,assume that F pHγq Ď Hα, and let Y : r0, T sˆΩ Ñ Hγ be a pFtqtPr0,T s/BpHγq-adaptedstochastic process which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s it holdsP-a.s. that

Yt “ 1tF pYzthq2Hα

ą1hu Yzth (8.70)

` 1tF pYzthq2αď1hu e

Apt´zthq

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

.

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated nonlinearity-stopped ex-ponential Euler approximation for the SPDE




8.2.2.2 Nonlinearity-stopped linear-implicit Euler method

Definition 8.2.13 (Nonlinearity-stopped linear-implicit Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochasticprocess which fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u itholds P-a.s. that

Yn`1 “ 1tF pYnq2HαąNTuYn (8.72)

` 1tF pYnq2HαďNTu

`

IdH ´TNA˘´1

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

.

Then we call Y a nonlinearity-stopped linear-implicit Euler approximation for theSPDE



Definition 8.2.14 (Naturally-interpolated nonlinearity-stopped linear-implicit Eu-ler approximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ´η, γs,assume that F pHγq Ď Hα, and let Y : r0, T sˆΩ Ñ Hγ be a pFtqtPr0,T s/BpHγq-adaptedstochastic processes which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s itholds P-a.s. that

Yt “ 1tF pYzthq2Hα

ą1hu Yzth ` 1tF pYzthq2Hα

ď1hu pIdH ´pt´ zthqAq´1

ˆ

Yzth

` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

. (8.74)

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated nonlinearity-stopped linear-implicit Euler approximation for the SPDE




8.2.3 Milstein type approximations for SPDEs

8.2.3.1 Exponential Milstein method

Definition 8.2.15 (Exponential Milstein approximations*). Assume the setting inSection 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is continuouslyFrechet differentiable, and let N P N. Then we say that Y is a pΩ,F ,P, pFtqtPr0,T sq-exponential Milstein approximation for the SPDE



is a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic processes which fulfills for all n Pt0, 1, . . . , N ´ 1u that Y0 “ ξ and


eATN

´

Yn ` F`

Yn˘

TN

¯ı

P,BpHγq`

żpn`1qTN

nTN

eATN B

`

Yn˘

dWs

`

żpn`1qTN

nTN

eATN B1

`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

.

(8.77)

Definition 8.2.16 (Naturally-interpolated exponential Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, and let h P p0,8q. Then we saythat Y is a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Milstein approxi-mation for the SPDE


with time step size h if and only if it holds that Y P Mpr0, T s ˆ Ω, Hγq is apFtqtPr0,T s/BpHγq-adapted stochastic process which fulfills for all t P p0, T s that Y0 “ ξand

rYtsP,BpHγq “

„

eApt´zthq

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq

P,BpHγq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.79)



be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Milstein approximationfor the SPDE



rYtsP,BpHγq “

„

eAtξ `

ż t

0

eApt´tsuhqF`

Ytsuh

˘

ds

P,BpHγq

`

ż t

0

eApt´tsuhq”

B`

Ytsuh

˘

`B1`

Ytsuh

˘s

∫tsuh

BpYtsuhq dWu

ı

dWs.

(8.81)

8.2.3.2 Linear-implicit Milstein method

Definition 8.2.17 (Linear-implicit Milstein approximations). Assume the settingin Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is contin-uously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be apFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic processes which fulfills Y0 “ ξ and whichfulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´TNA˘´1

˜

Yn ` F`

Yn˘

TN`

żpn`1qTN

nTN

B`

Yn˘

dWs

`

żpn`1qTN

nTN

B1`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

.

(8.82)

Then we call Y a linear-implicit Milstein approximation for the SPDE




Definition 8.2.18 (Naturally-interpolated linear-implicit Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, and let Y : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s/BpHγq-adapted stochastic processes which fulfills Y0 “ ξand which fulfills that for all t P p0, T s it holds P-a.s. that

Yt “pIdH ´pt´ zthqAq´1

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.84)

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Milsteinapproximation for the SPDE



Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ bea pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Milstein approximationfor the SPDE



Yt “`


IdH ´hA˘´ttuhh

ξ

`

ż t

0

`



F`

Ytsuh

˘

ds

`

ż t

0

`



B`

Ytsuh

˘

dWs

`

ż t

0

`



”

B1`

Yttuh

˘s

∫ttuh

B`

Yttuh

˘

dWu

ı

dWs.

(8.87)


8.2.3.3 Linear-implicit Crank-Nicolson-Milstein method

Definition 8.2.19 (Linear-implicit Crank-Nicolson-Milstein approximations). As-sume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic processes which fulfillsY0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´T

2NA˘´1

˜

`

IdH `T

2NA˘

Yn ` F`

Yn˘

TN`

żpn`1qTN

nTN

B`

Yn˘

dWs

`

żpn`1qTN

nTN

B1`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

. (8.88)

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Crank-Nicolson-Milstein approximation for the SPDE



Definition 8.2.20 (Naturally-interpolated linear-implicit Crank-Nicolson-Milsteinapproximations). Assume the setting in Section 7.1.2, assume that γ “ β, assumethat B : Hγ Ñ HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, andlet Y : r0, T s ˆ Ω Ñ Hγ be a pFtqtPr0,T s/BpHγq-adapted stochastic processes whichfulfills Y0 “ ξ and which fulfills that for all t P p0, T s it holds P-a.s. that

Yt “`

IdH ´12pt´ zthqA

˘´1

ˆ

Yzth `12AYzth pt´ zthq ` F

`

Yzth

˘

pt´ zthq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.90)

Then we call Y a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Crank-Nicolson-Milstein approximation for the SPDE




Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ bea pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated linear-implicit Crank-Nicolson-Milsteinapproximation for the SPDE



Yt “`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘´ttuhh

ξ

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


”

12AYtsuh ` F

`

Ytsuh

˘

ı

ds

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


B`

Ytsuh

˘

dWs

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


”

B1`

Yttuh

˘s

∫ttuh

B`

Yttuh

˘

dWu

ı

dWs.

(8.93)

8.2.4 Strong convergence analysis for exponential Euler ap-proximations

In this subsection we establish strong convergence with suitable rates of convergenceof exponential Euler approximations; see Definition 8.2.1. In this subsection wemainly follow the analysis in Kurniawan [16].

Lemma 8.2.21 (Regularity for the numerical approximations*). Assume the settingin Subsection 7.1.2, let N P N, and let Y : r0, T sˆΩ Ñ Hγ be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation for the SPDE


with time step size TN. Then

suptPr0,T s

YtLppP;¨Hγ qď max

1,?

2 ξLppP;¨Hγ q

(

(8.95)

¨ E1´η

„

T 1´η?

2 F LippHγ,Hγ´ηq?1´η

à

T 1´η p pp´ 1q BLippHγ ,HSpU,Hγ´η2qq

ă 8.


Proof of Lemma 8.2.21*. Theorem 4.8.5 and Holder’s inequality imply that for allt P r0, T s it holds that

YtLppP;¨Hγ qď ξLppP;¨Hγ q

` F LippHγ ,Hγ´ηq

„

tp1´ηq

p1´ηq

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

` BLippHγ ,HSpU,Hγ´η2qq

„

ppp´1q2

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

.

(8.96)


YtLppP;¨Hγ qď ξLppP;¨Hγ q

`

„ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

¨

„

F LippHγ ,Hγ´ηqTp1´ηq2?

1´η` BLippHγ ,HSpU,Hγ´η2qq

?ppp´1q?

2

.

(8.97)

Induction hence proves that for all t P r0, T s it holds that YtLppP;¨Hγ qă 8.

Moreover, combining (8.97) with the estimate that for all a, b P R it holds thatpa` bq2 ď 2a2 ` 2b2 shows that for all t P r0, T s it holds that

max

1, Yt2LppP;¨Hγ q

(

ď max

1, 2 ξ2LppP;¨Hγ q

(

`

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

¨

”

F LippHγ ,Hγ´ηq

?2T

p1´ηq2?


a

p pp´ 1qı2

.

(8.98)


Next note that for all u P r0, T s it holds that

suptPr0,us

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

ď suptPr0,us

ż t

0

pt´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds

“ suptPr0,us

ż u

u´t

pt´ rs´ pu´ tqsq´η”

supvPr0,s´pu´tqs max

1, Yv2LppP;¨Hγ q

(

ı

ds

“ suptPr0,us

ż u

u´t

pu´ sq´η”

supvPr0,s`tús max

1, Yv2LppP;¨Hγ q

(

ı

ds

ď suptPr0,us

ż u

u´t

pu´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds

“

ż u

0

pu´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds.

(8.99)

Putting this into (8.98) proves that for all u P r0, T s it holds that

suptPr0,us max

1, Yt2LppP;¨Hγ q

(

ď max

1, 2 ξ2LppP;¨Hγ q

(

`

ż u

0

pu´ sq´η”

suptPr0,ss max

1, Yt2LppP;¨Hγ q

(

ı

ds

¨

”

F LippHγ ,Hγ´ηq

?2T

p1´ηq2?


a

p pp´ 1qı2

.

(8.100)

Combining this with Corollary 1.4.6 completes the proof of Lemma 8.2.21.


Lemma 8.2.22 (More regularity for the numerical approximation processes*). As-sume the setting in Subsection 7.1.2, let N P N, and let Y : r0, T s ˆ Ω Ñ Hγ bea pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation for theSPDE


with time step size TN. Then

(i) it holds for all t P r0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Yt ´ eAtY0 P

Hr

˘

“ 1 and

›

›Yt ´ eAtY0

›

›

LppP;¨Hr qď max

#

1, supnPt0,1,...,Nu

YnTNLppP;¨Hγ q

+

¨

«

F LippHγ ,Hγ´ηqtp1`γ´η´rq

p1` γ ´ η ´ rq`

a

p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq

p2` 4β ´ 4rq12

ff

ă 8

(8.102)

and

(ii) it holds for all t P p0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Yt P Hr

˘

“ 1and

YtLppP;¨Hr qď

X0LppP;¨Hγ q

tpr´γq`max

!

1, supnPt0,1,...,Nu YnTNLppP;¨Hγ q

)

¨

„

F LippHγ,Hγ´ηqtp1`γ´η´rq

p1`γ´η´rq`

?p pp´1q BLippHγ,HSpU,Hβqq

tp12`β´rq

p2`4β´4rq12

ă 8.

(8.103)

Proof of Lemma 8.2.22*. First of all, recall that for all t P r0, T s it holds that

“

Yt ´ eAtY0

‰

P,BpHγq

“

„ż t

0

eApt´tsuT N qF pYtsuT Nq ds

P,BpHγq`

ż t

0

eApt´tsuT N qBpYtsuT Nq dWs.

(8.104)

Moreover, note that Theorem 4.8.5 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηq it


holds that

ż t

0

›

›

›eApt´tsuT N qF pYtsuT N

q

›

›

›

LppP;¨Hr qds

ď F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

„ż t

0

pt´ sqpγ´η´rq ds

“ F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

tp1`γ´η´rq

p1` γ ´ η ´ rqă 8.

(8.105)

Furthermore, observe that Theorem 4.8.5 ensures that for all t P r0, T s, r P rγ, β`12q

it holds that

„ż t

0

›

›

›eApt´tsuT N qBpYtsuT N

q

›

›

›

2

LppP;¨HSpU,Hrqqds

12


#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

„ż t

0

pt´ sqp2β´2rq ds

12


#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

tp12`β´rq

p1` 2β ´ 2rq12ă 8.

(8.106)

Combining (8.104), (8.105), and (8.106) with Theorem 4.8.5 and Theorem 6.3.29completes the proof of Lemma 8.2.22.

Lemma 8.2.23 (Regularity of a time integral associated to the numerical approx-imations*). Assume the setting in Section 7.1.2, let ρ P r0, 1q, q P r1,8q, and forevery N P N let Y N : r0, T sˆΩ Ñ Hγ be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolatedexponential Euler approximation for the SPDE


with time step size TN. Then it holds for all r P p´8,mintp1´ρqq, 1´ η, 12` β ´ γuqthat

supNPN

suptPr0,T s

„

N rt

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ă 8. (8.108)


Proof of Lemma 8.2.23*. First of all, observe that for all N P N it holds that

suptPr0,TNs

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 suptPp0,T s›

›Y Nt

›

›

LppP;¨Hγ qT p1´ρqq

p1´ ρq1qN p1´ρqq.

(8.109)

Next note that for all r P rγ,mint1 ` γ ´ η, 12 ` βuq, N P N, t P rTN, T s it holdsthat

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď 2 supsPp0,T s

›

›Y Ns

›

›

LppP;¨Hγ q

„

TN

∫0pt´ sq´ρ ds

1q

`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff«

t

∫TN

pt´ sq´ρ

`

tsuT N˘q pr´γq

ds

ff1q

ď 2 supsPp0,T s

›

›Y Ns

›

›

LppP;¨Hγ q

T p1´ρqq

N p1´ρqq p1´ ρq1q

`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

„

t´TN

∫0

pt´ TN ´ sq´ρ

sq pr´γqds

1q

.

(8.110)

This implies that that for all r P rγ,mint1` γ´ η, 12`βuq, N P N, t P rTN, T s withρ` qpr ´ γq ď 1 it holds that

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 supsPp0,T s›

›Y Ns

›

›



`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

¨ rt´ TNsr1´ρq´pr´γqs

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

.

(8.111)

Combining this with (8.109) proves that for all r P rγ,mintγ`p1´ρqq, γ`1´η, 12`βuq,


N P N it holds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 supsPp0,T s›

›Y Ns

›

›



`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

¨ T rγ`p1´ρqq´rs

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

.

(8.112)

In the next step we observe that Theorem 6.3.29, Lemma 4.8.6 and Lemma 8.2.22ensure that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holds that

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qď

›

›

›

`

eAps´tsuT N q ´ IdH˘

Y NtsuT N

›

›

›

LppP;¨Hγ q

`

ż s

tsuT N

›

›eAps´tuuT N qF pY NtuuT N

q›

›

LppP;¨Hγ qdu

`

«

p pp´ 1q

2

ż s

tsuT N

›

›

›eAps´tuuT N qBpY N

tuuT Nq

›

›

›

2

LppP;¨HSpU,Hγ qqdu

ff12

ď`

s´ tsuT N˘pr´γq

›

›

›Y N

tsuT N

›

›

›

LppP;¨Hr q

`

ż s

tsuT N

ps´ uq´η›

›

›F pY N

tuuT Nq

›

›

›

LppP;¨Hγ´ηqdu

`

«

p pp´ 1q

2

ż s

tsuT N

ps´ uqp2β´2γq›

›

›BpY N

tuuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqdu

ff12

ă 8.

(8.113)

This implies that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holdsthat›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qď

„

T

N

pr´γqˇ

ˇtsuT Nˇ

ˇ

pγ´rq

«

supuPp0,T s

upr´γq›

›Y Nu

›

›

LppP;¨Hr q

ff

`“

TN

‰p1´ηq F LippHγ,Hγ´ηq

p1´ηqmax

"

1, supnPt1,2,...,Nu›

›Y NnTN

›

›

LppP;¨Hγ q

*

`“

TN

‰p12`β´γq BLippHγ,HSpU,Hβqq

p1`2β´2γq12

”

p pp´1q2

ı12

max

"

1, supnPt1,2,...,Nu›

›Y NnTN

›

›

LppP;¨Hγ q

*

.

(8.114)


This shows that for all N P N, r P rγ,mint1` γ ´ η, 12` βuq it holds that

supsPrTN,T s

„

ˇ

ˇtsuT Nˇ

ˇ

pr´γq›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ď

”

“

TN

‰pr´γq`“

TN

‰p1´ηq Tpr´γq F LippHγ,Hγ´ηq

p1´ηq

`“

TN

‰p12`β´γq Tpr´γq BLippHγ,HSpU,Hβqq

p12`β´γq12

”

p pp´1q4

ı12 ı

¨max

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.115)

Putting this into (8.112) proves that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´η, 12` βuq it holds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

«

2T p1´ρqq


`

”

1`T p1´ηq F LippHγ,Hγ´ηq

p1´ηq`

T p12`β´γq BLippHγ,HSpU,Hβqq

p12`β´γq12

”

p pp´1q4

ı12 ı

¨T p1´ρqq

N pr´γq

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

ff

¨max

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.116)

Hence, we obtain that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´ η, 12 ` βuq itholds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

„

52` F LippHγ ,Hγ´ηq ` BLippHγ ,HSpU,Hβqq

?p pp´1q

2

¨

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

maxpT 2, 1q

mint1´ ρ, 1´ η, 12` β ´ γuN pr´γqmax

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.117)


In the next result, Corollary 8.2.24, an estimate for the strong approximationerror of exponential Euler approximations is presented.


Corollary 8.2.24 (*). Assume the setting in Section 7.1.2, let X : r0, T s ˆΩ Ñ Hγ

be a stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ qă 8 and which is a

pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


let N P N, and let Y : r0, T sˆΩ Ñ Hγ be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolatedexponential Euler approximation for the SPDE (8.118) with time step size TN. Then

suptPr0,T s

Xt ´ YtLppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à


¨?

2 suptPr0,T s

«

ż t

0

›

›

›eApt´sq

”

F pYsq ´ eAps´tsuT N qF pYtsuT N

q

ı›

›

›

LppP;¨Hγ qds

`

„

p pp´1q2

t

∫0

›

›

›eApt´sq

”

BpYsq ´ eAps´tsuT N qBpYtsuT N

q

ı›

›

›

2


12ff

ă 8.

(8.119)

Corollary 8.2.24 is an immediate consequence of Proposition 7.1.6 and of Theo-rem 6.3.29.

Theorem 8.2.25 (*). Assume the setting in Section 7.1.2, let X : r0, T s ˆ Ω Ñ Hγ

be a stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ qă 8 and which is a

pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


and for every N P N let Y N : r0, T s ˆ Ω Ñ Hγ be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponential Euler approximation for the SPDE (8.120) with time stepsize TN. Then it holds for all r P

`

´8,mint1´ η, 12` β ´ γu˘

that

supNPN

suptPr0,T s

„

N r›

›Xt ´ YNt

›

›

LppP;¨Hγ q

ă 8. (8.121)


Proof of Theorem 8.2.25*. Observe that Theorem 4.8.5 and Lemma 4.8.6 imply thatfor all t P r0, T s, ε P p0, 1´ ηq it holds that

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ e

Aps´tsuT N qF pY NtsuT N

q

ı›

›

›

LppP;¨Hγ qds

ď

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ F pY

NtsuT N

q

ı›

›

›

LppP;¨Hγ qds

`

ż t

0

›

›

›eApt´sq

`

IdH éAps´tsuT N q

˘

F pY NtsuT N

q

›

›

›

LppP;¨Hγ qds

ď

ż t

0

pt´ sq´η›

›

›F pY N

s q ´ F pYN

tsuT Nq

›

›

›

LppP;¨Hγ´η qds

`

ż t

0

pt´ sq´η´ε`

s´ tsuT N˘ε›

›

›F pY N

tsuT Nq

›

›

›

LppP;¨Hγ´η qds.

(8.122)

This ensures that for all t P r0, T s, ε P p0, 1´ ηq it holds that

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ e

Aps´tsuT N qF pY NtsuT N

q

ı›

›

›

LppP;¨Hγ qds


ż t

0

pt´ sq´η›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qds

`

„

T

N

ε«

supsPr0,T s

›

›

›F pY N

tsuT Nq

›

›

›

LppP;¨Hγ´η q

ff

T p1´η´εq

p1´ η ´ εq.

(8.123)

Furthermore, Theorem 4.8.5 and Lemma 4.8.6 prove that for all t P r0, T s, ε P


p0, 12` β ´ γq it holds that

„

t

∫0

›

›

›eApt´sq

”

BpY Ns q ´ e

Aps´tsuT N qBpY NtsuT N

q

ı›

›

›

2


12

ď

«

t

∫0pt´ sqp2β´2γq

›

›

›BpY N

s q ´BpYN

tsuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqds

ff12

`

«

t

∫0pt´ sqp2β´2γ´2εq

`

s´ tsuT N˘2ε

›

›

›BpY N

tsuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqds

ff12

ď |B|C0,1pHγ ,HSpU,Hβqq

„

t

∫0pt´ sqp2β´2γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

2

LppP;¨Hγ qds

12

`

„

T

N

ε«

supsPr0,T s

›

›

›BpY N

tsuT Nq

›

›

›

LppP;¨HSpU,Hβqq

ff

T p12`β´γ´εq

p1` 2β ´ 2γ ´ 2εq12.

(8.124)

Combining (8.123), (8.124), Corollary 8.2.24, and Lemma 8.2.23 completes the proofof Theorem 8.2.25.

Class exercise 8.2.26 (Convergence speed of exponential Euler approximations*).Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let pWtqtPr0,T s be an IdH-

cylindrical pΩ,F ,P, pFtqtPr0,T sq-Wiener process, let b : p0, 1qˆRÑ R be a globally Lip-schitz continuous function, let ξ P H, let X : r0, T sˆΩ Ñ H be a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq

(8.125)for x P p0, 1q, t P r0, T s, assume that suptPr0,T sE

“

Xt2H

‰

ă 8, and for every N P N

let Y N : r0, T s ˆ Ω Ñ H be a pΩ,F ,P, pFtqtPr0,T sq-naturally-interpolated exponentialEuler approximation for the SPDE (8.125) with time step size TN.

(i) For which r P R does it holds that there exist a real number C P R such thatfor all N P N it holds that E

“

XT ´ YNT H

‰

ď CN´r?

(ii) For which r P R does it holds that for every p P p0,8q there exist a real numberC P R such that for all N P N it holds that suptPr0,T s

›

›Xt ´ YNt

›

›

LppP;¨Hqď

CN´r?


8.3 Noise approximations for SPDEs

8.3.1 Noise perturbation estimates

The next result, Corollary 8.3.1, is an immediate consequence of Proposition 7.1.6,Theorem 4.8.5, and Theorem 6.3.29.

Corollary 8.3.1 (Noise perturbation). Assume the setting in Section 7.1.2, let θ Prγ ´ β, 12q, B P C0,1pHγ, HSpU,Hβqq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochasticprocesses which satisfy supsPr0,T s

“

XsLppP;¨Hγ q` XsLppP;¨Hγ q

‰

ă 8, which satisfy

that X is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE


and which satisfy that X is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXt ““

AXt ` F pXtq‰

dt` BpXtq dWt, t P r0, T s, X0 “ ξ. (8.127)

Then

suptPr0,T s

›

›Xt ´ Xt

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à


¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPp0,T q

›

›BpXsq ´ BpXsq›

›

LppP;¨HSpU,Hγ´θqq

ff

ă 8.

(8.128)

Proof. Proposition 7.1.6 ensures that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à


¨?

2 suptPr0,T s

›

›

›

›

„

Xt ´

ż t

0

eApt´sqF pXsq ds´

ż t

0

eApt´sqBpXsq dWs

`

„ż t

0

eApt´sqF pXsq ds`

ż t

0

eApt´sqBpXsq dWs ´ Xt

›

›

›

›

LppP;¨Hγ q

ă 8.

(8.129)

8.3. NOISE APPROXIMATIONS FOR SPDES 289

This implies that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à


¨?

2 suptPr0,T s

›

›

›

›

ż t

0

eApt´sq“

BpXsq ´ BpXsq‰

dWs

›

›

›

›

LppP;¨Hγ q

ă 8.

(8.130)

Theorem 6.3.29. and Theorem 4.8.5 hence prove that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


à


¨?

2

«

p pp´1q2

suptPr0,T s

ż t

0

s´2θds

ff12 «

supsPp0,T q

›

›BpXsq ´ BpXsq›

›

LppP;¨HSpU,Hγ´θqq

ff

ă 8.

(8.131)

This completes the proof of Corollary 8.3.1.

8.3.2 Noise approximations for SPDEs

The next result, Corollary 8.3.2, is an immediate consequence from Corollary 8.3.1above.


Corollary 8.3.2 (Noise discretizations). Assume the setting in Section 7.1.2, letθ P rγ ´ β, 12q, R, R P LpUq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochastic processeswhich satisfy supsPr0,T s

“

XsLppP;¨Hγ q` XsLppP;¨Hγ q

‰

ă 8, which satisfy that X is

a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtqRdWt, t P r0, T s, X0 “ ξ, (8.132)

and which satisfy that X is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXt ““

AXt ` F pXtq‰

dt`BpXtq R dWt, t P r0, T s, X0 “ ξ. (8.133)

Then

suptPr0,T s

›

›Xt ´ Xt

›

›

LppP;¨Hγ qď

«

supvPHγ

BpvqrR ´ RsHSpU,Hγ´θq

maxt1, vHγu

ff

¨ Ep1´ηq„

T 1´η?


à

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqRLpUq

¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPr0,T s

›

›maxt1, XsHγu›

›

LppP;|¨|q

ff

ă 8.

(8.134)

The next result, Corollary 8.3.3, illustrates how strong convergence rates for noisediscretizations can be obtained.

8.3. NOISE APPROXIMATIONS FOR SPDES 291

Corollary 8.3.3. Assume the setting in Section 7.1.2, let θ P rγ ´ β, 12q, r P

r0,8q, pRNqNPN0 Ď LpUq satisfy supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq

maxt1,vHγ uă 8, and

let XN : r0, T s ˆ Ω Ñ Hγ, N P N0, be stochastic processes with the property that@N P N0 : supsPr0,T s XsLppP;¨Hγ q

ă 8 and with the property that for all N P N0 it

holds that XN is a pΩ,F ,P, pFtqtPr0,T sq-mild solution of the SPDE

dXNt “

“

AXNt ` F pX

Nt q

‰

dt`BpXNt qRN dWt, t P r0, T s, X0 “ ξ. (8.135)

Then

suptPr0,T s

›

›X0t ´X

Nt

›

›

LppP;¨Hγ qď

1

N r

«

supMPN

supvPHγ

M rBpvqrR0 ´RM sHSpU,Hγ´θq

maxt1, vHγu

ff

¨ Ep1´ηq„

T 1´η?


à

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqR0LpUq

¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPr0,T s

supMPN

›

›maxt1, XMs Hγu

›

›

LppP;|¨|q

ff

ă 8.

(8.136)

Proof of Corollary 8.3.3. First, observe that Proposition 7.1.15 ensures that for allM P N0 it holds that

suptPr0,T s

›

›max

1, XMt Hγ

(›

›

LppP;|¨|qď?

2›

›max

1, ξHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


à

T 1´ηppp´ 1q Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq

ă 8.

(8.137)

Next note that the assumption that supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq

maxt1,vHγ uă 8

implies that

supMPN0

Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq“ sup

MPN0

supvPHγ

«

BpvqRMHSpU,Hγ´η2q

max

1, vHγ(

ff

ď supvPHγ

«

BpvqHSpU,Hγ´η2qR0LpUq

max

1, vHγ(

ff

` supMPN0

supvPHγ

«

BpvqrR0 ´RM sHSpU,Hγ´η2q

max

1, vHγ(

ff

ă 8.

(8.138)


This and (8.137) prove that

supMPN0

suptPr0,T s

›

›max

1, XMt Hγ

(›

›

LppP;|¨|qă 8. (8.139)

This and Corollary 8.3.2 complete the proof of Corollary 8.3.3.

8.4 Full discretizations for SPDEs

8.4.1 Setting

Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letU Ď U be an orthonormal basis of U , let λ : B Ñ R be a function, assume thatDpAq “

v P H :ř

bPB |λb 〈b, v〉H |2ă 8

(

, assume that for all v P DpAq it holds thatAv “

ř

bPB λb 〈b, v〉H b and let pπIqIPPpBq Ď LpHγ´ηq and p$IqIPPpUq Ď LpUq satisfythat for all v P Hγ´η, u P U , I P PpBq, J P PpUq it holds that

πIpvq “ÿ

bPI

〈b, v〉H b and $Jpuq “ÿ

bPJ

〈b, u〉U b. (8.140)

8.4.2 Full-discrete spectral Galerkin exponential Euler methodfor SPDEs

Definition 8.4.1 (Full discrete spectral Galerkin exponential Euler approxima-tions). Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq andlet Y : t0, 1, . . . , Nu ˆ Ω Ñ πIpHγq be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochasticprocess which fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u itholds P-a.s. that

Yn`1 “ eATN

˜

Yn ` πIpF pYnqqTN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

. (8.141)

Then we call Y a full-discrete spectral Galerkin exponential Euler approximation forthe SPDE


with time step size TN, spatial approximation I and noise approximation J .

8.4. FULL DISCRETIZATIONS FOR SPDES 293

1 function Y = ExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;

10 end

Matlab code 8.1: A Matlab function for simulating a full-discrete spectralGalerkin exponential Euler approximation for the SPDE (7.78).

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = ExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = ExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = ExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.2: A Matlab code for simulating a full-discrete spectral Galerkinexponential Euler approximation for an SPDE of the form (7.78).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 8.1: Result of a call of the Matlab code 8.2.


8.4.3 Full-discrete spectral Galerkin linear-implicit Euler methodfor SPDEs

Definition 8.4.2 (Full-discrete spectral Galerkin linear-implicit Euler approxima-tions). Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq, andlet Y : t0, 1, . . . , Nu ˆ Ω Ñ πIpHγq be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochasticprocess which fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u itholds P-a.s. that

Yn`1 “`

IdH ´TNA˘´1

˜

Yn ` πI`

F pYnq˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

(8.143)

Then we call Y a full-discrete spectral Galerkin linear-implicit Euler approximationfor the SPDE



1 function Y = LinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;

10 end

Matlab code 8.3: A Matlab function for simulating a full-discrete spectralGalerkin linear-implicit Euler approximation of the SPDE (7.78).

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;


6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = LinImpEuler (T, v , f , b , xi ,N,M)9 plot ( Preimage ,Y) ;

10 Y = LinImpEuler (T, v , f , b , xi ,N,M)11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = LinImpEuler (T, v , f , b , xi ,N,M)13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.4: A Matlab code for simulating a full-discrete spectral Galerkinlinear-implicit Euler approximation of the SPDE (7.78).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1



8.4.4 Full-discrete spectral Galerkin nonlinearity-stopped ex-ponential Euler method for SPDEs

Definition 8.4.3 (Full-discrete spectral Galerkin nonlinearity-stopped exponentialEuler approximations). Assume the setting in Section 8.4.1, let N P N, I P PpBq, J PPpUq, α P rγ´η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , NuˆΩ Ñ πIpHγq

be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic process which fulfills Y0 “ πIpξq andwhich fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.145)

` 1tπIpF pYnqq2HαďNTueA

TN

˜

Yn ` πI`

F pYnq˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

Then we call Y a full-discrete spectral Galerkin nonlinearity-stopped exponential Eu-ler approximation for the SPDE



1 function Y = StopExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;

10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end

Matlab code 8.5: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped exponential Euler approximation for a generalized ver-sion of the SPDE (7.78) with γ P p1

5, 1

4q and α “ 0.

1 clear a l l2 rng ( ’ d e f a u l t ’ )


3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.6: A Matlab code for simulating a full-discrete spectral Galerkinnonlinearity-stopped exponential Euler approximation for a generalized version ofthe SPDE (7.78) with γ P p1

5, 1

4q and α “ 0.

8.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicit Euler method for SPDEs

Definition 8.4.4 (Full-discrete spectral Galerkin nonlinearity-stopped linear-im-plicit Euler approximations). Assume the setting in Section 8.4.1, let N P N, I PPpHq, J P PpUq, α P rγ´ η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , NuˆΩ Ñ πIpHγq be a pFnTNqnPt0,1,...,Nu/BpHγq-adapted stochastic process which fulfillsY0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.147)

` 1tπIpF pYnqq2HαďNTu

`

IdH ´TNA˘´1

˜

Yn ` πI`

F`

Yn˘˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

Then we call Y a spectral Galerkin nonlinearity-stopped linear-implicit Euler approx-imation for the SPDE




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4



1 function Y = StopLinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;

10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end

Matlab code 8.7: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped linear-implicit Euler approximation for a generalizedversion of the SPDE (7.78) with γ P p1

5, 1

4q and α “ 0.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.8: A Matlab code for simulating a spectral Galerkin nonlinearity-stopped linear-implicit Euler approximation for a generalized version of theSPDE (7.78) with γ P p1

5, 1

4q and α “ 0.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4



Chapter 9

Solutions to selected exercises

9.1 Chapter 2

9.1.1 Solution to Exercise 2.2.6

Lemma 9.1.1. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space, andlet f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if it holdsfor all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.

Proof of Lemma 9.1.1. First of all, recall that every ϕ P CpE,Rq is BpEq/BpRq-measurable. This shows that f PMpF ,BpEqq implies that for every ϕ P CpE,Rq itholds that the composition ϕ ˝ f is F/BpRq-measurable. It thus remains to provethat @ϕ P CpE,Rq : ϕ ˝ f P MpF ,BpRqq ensures that f is F/BpEq-measurable.This, in turn, is an immediate consequence from Proposition 2.2.5. The proof ofLemma 9.1.1 is thus completed.

303

304 CHAPTER 9. SOLUTIONS TO SELECTED EXERCISES

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