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Southern Illinois University CarbondaleOpenSIUCMiscellaneous (presentations, translations,interviews, etc) Department of Mathematics
1996
Stochastic Differential Systems with Memory:Theory, Examples and Applications (SixthWorkshop on Stochastic Analysis)Salah-Eldin A. MohammedSouthern Illinois University Carbondale, [email protected]
Follow this and additional works at: http://opensiuc.lib.siu.edu/math_miscPart of the Mathematics Commons
Sixth Workshop on Stochastic Analysis; July 29 - August 4, 1996; Geilo, Norway. Paper forproceedings is posted at http://opensiuc.lib.siu.edu/math_articles/50/.
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Recommended CitationMohammed, Salah-Eldin A., "Stochastic Differential Systems with Memory: Theory, Examples and Applications (Sixth Workshop onStochastic Analysis)" (1996). Miscellaneous (presentations, translations, interviews, etc). Paper 28.http://opensiuc.lib.siu.edu/math_misc/28
STOCHASTIC DIFFERENTIAL SYSTEMS
WITH MEMORY
THEORY, EXAMPLES AND APPLICATIONS
Geilo, Norway : July 29-August 4, 1996
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
Outline of Lectures
Lecture I. Existence.
1. Simple examples: The noisy feedback loop: dx(t) = x(t− r) dW (t).Solution process x(t) is not a Markov process in R. No closedform solution when r > 0. Compare with the case r = 0. Thelogistic time-lag model with Gaussian noise:
dx(t) = [α− βx(t− r)]x(t) dt + σx(t) dW (t).
The classical “heat-bath” model of R. Kubo: Motion of a largemolecule in a viscous fluid.
2. General Formulation. Choice of state space. Pathwise existenceand uniquenss of solutions to sfde’s under local Lipschitz andlinear growth hypotheses on the coefficients. Existence theoremallows for stochastic white-noise perturbations of the memory,e.g.
dx(t) =∫
[−r,0]
x(t + s) dW (s)
dW (t) t > 0
Above sfde is not covered by classical results of Protter, Metivierand Pellaumail, Doleans-Dade.
3. Mean Lipschitz, smooth and sublinear dependence of the tra-jectory random field.
2
Lecture II. Markov Behavior and the Generator.
1. Markov (Feller) property holds for the trajectory random field.Time homogeneity.
2. Construction of the semigroup. Semigroup is not strongly con-tinuous for positive delay. Domain of strong continuity does notcontain tame (or cylinder) functions with evaluations away from0, but contains “quasitame” functions. These are weakly densein the underlying space of continuous functions and generate theBorel σ-algebra of the state space.
3. Derivation of a formula for the weak infinitesimal generator ofthe semigroup for sufficiently regular functions, and for a largeclass of quasitame functions.
3
Lecture III. Regularity. Classification of SFDE’s.
1. Pathwise regularity of the trajectory random field in the timevariable. α-Holder continuity.
2. Almost sure (pathwise) dependence on the initial state. Non-existence of the stochastic flow for the singular sdde dx(t) =x(t − r) dW (t). Breakdown of linearity and local boundedness.Classification of sfde’s into regular and singular types.
3. Results on sufficient conditions for regularity of linear systemsdriven by white noise or semimartingales.
4. Sussman-Doss type nonlinear sfde’s. Existence and compactnessof semiflow.
5. Path regularity of general non-linear sfde’s with “smooth mem-ory”.
4
Lecture IV. Ergodic Theory of Linear SFDE’s.
1. Existence of stochastic semiflows for certain classes of linearsfde’s with smooth memory terms. The cocycle and its perfec-tion.
2. Compactness of the semiflow in the finite memory case.3. Ruelle-Oseledec multiplicative ergodic theorem in Hilbert space.
Existence of a discrete Lyapunov spectrum. The Stable Mani-fold Theorem (viz. random saddles) for hyperbolic linear sfde’sdriven by white noise. The case of helix noise.
5
Lecture V. Stability. Examples and Case Studies.
1. Estimates on the maximal exponential growth rate for the sin-gular noisy feedback loop: dx(t) = σx(t − r) dW (t). Stability andinstability for small σ (or large r) using a Lyapunov functionalargument. Comparison with the non-delay case for large σ.
2. Derivation of estimates on the top Lyapunov exponent λ1 for var-ious examples of one-dimensional regular sfde’s driven by whitenoise or a martingale with stationary ergodic increments.
3. Lyapunov spectrum for sdde dx(t) = x((t− 1)−) dN(t) driven by aPoisson process N . Characterization of the Lyapunov spectrum.
6
Lecture VI. Miscellany
1. Malliavin Calculus of SFDE’s. Regularity of the solution x(t, ω)in ω. Malliavin smoothness and existence of smooth densities.Classical solution of a degenerate parabolic pde as an applica-tion.
2. Small delays. Applications to sode’s. A proof of the classicalexistence theorem for solutions to sode’s.
3. Affine systems of sfde’s. Lyapunov spectrum. The hyperbolicsplitting. Existence of stationary solutions in the hyperboliccase. Application to simple population model.
4. Random delays. Induced measure-valued process. Random fam-ilies of Markov fields and random generators.
5. Infinite memory and stationary solutions.
7
References
[AKO] Arnold, L., Kliemann, W. and Oeljeklaus, E. Lyapunov expo-nents of linear stochastic systems, in Lyapunov Exponents, SpringerLecture Notes in Mathematics 1186 (1989), 85–125.
[AOP] Arnold, L. Oeljeklaus, E. and Pardoux, E., Almost sure andmoment stability for linear Ito equations, in Lyapunov Exponents,Springer Lecture Notes in Mathematics 1186 (ed. L. Arnoldand V. Wihstutz) (1986), 129–159.
[B] Baxendale, P.H., Moment stability and large deviations for linear stochas-tic differential equations, in Ikeda, N. (ed.) Proceedings of the TaniguchiSymposium on Probabilistic Methods in Mathematical Physics, Katata andKyoto (1985), 31–54, Tokyo: Kinokuniya (1987).
[CJPS] Cinlar, E., Jacod, J., Protter, P. and Sharpe, M. Semimartingalesand Markov processes, Z. Wahrsch. Verw. Gebiete 54 (1980), 161–219.
[FS] Flandoli, F. and Schaumloffel, K.-U. Stochastic parabolic equationsin bounded domains: Random evolution operator and Lyapunov exponents,Stochastics and Stochastic Reports 29, 4 (1990), 461–485.
[M1] Mohammed, S.-E.A. Stochastic Functional Differential Equations, Re-search Notes in Mathematics 99, Pitman Advanced PublishingProgram, Boston-London-Melbourne (1984).
[M2] Mohammed, S.-E.A. Non-linear flows for linear stochastic delay equa-tions, Stochastics 17, 3 (1986), 207–212.
[M3] Mohammed, S.-E.A. The Lyapunov spectrum and stable manifolds forstochastic linear delay equations, Stochastics and Stochastic Reports29 (1990), 89–131.
8
[M4] Mohammed, S.-E.A. Lyapunov exponents and stochastic flows of linearand affine hereditary systems, (1992) (Survey article), Birkhauser(1992), 141–169.
[MS] Mohammed, S.-E.A. and Scheutzow, M.K.R. Lyapunov exponents oflinear stochastic functional differential equation driven by semimartingales.Part I: The multiplicative ergodic theory, (preprint, 1990), AIHP (toappear).
[P] Protter, Ph.E. Semimartingales and measure-preserving flows,Ann. Inst. Henri Poincare, Probabilites et Statistiques, vol. 22, (1986),127-147.
[R] Ruelle, D. Characteristic exponents and invariant manifolds inHilbert space, Annals of Mathematics 115 (1982), 243–290.
[S] Scheutzow, M.K.R. Stationary and Periodic Stochastic Differential Sys-tems: A study of Qualitative Changes with Respect to the Noise Level andAsymptotics, Habiltationsschrift, University of Kaiserslautern, W.Germany (1988).
[Sc] Schwartz, L., Radon Measures on Arbitrary Topological Spaces and Cylin-drical measures, Tata Institute of Fundamental Research, OxfordUniversity Press, (1973).
[Sk] Skorohod, A. V., Random Linear Operators, D. Reidel PublishingCompany (1984).
[SM] de Sam Lazaro, J. and Meyer, P.A. Questions de theorie desflots, Seminaire de Probab. IX, Springer Lecture Notes in Mathe-matics 465, (1975), 1–96.
9
[E] Elworthy, K. D.,Stochastic differential equations on manifolds,Cambridge (Cambridgeshire) ; New York : Cambridge Univer-sity Press, (18), 326 p. ; 23 cm. 1982, London MathematicalSociety lecture note series ; 70.
[D] Dudley, R.M., The sizes of compact subsets of Hilbert space andcontinuity of Gaussian processes, J. Functional Analysis,1, (1967),290-330.
[FS] Flandoli, F. and Schaumloffel, K.-U. Stochastic parabolic equa-tions in bounded domains: Random evolution operator and Lya-punov exponents, Stochastics and Stochastic Reports 29, 4 (1990),461–485.
[H] Hale, J.K., Theory of Functional Differential Equations, Springer-Verlag,New York, Heidelberg, Berlin, (1977).
[Ha] Has’minskii, R. Z., Stochastic Stability of Differential Equations, Si-jthoff & Noordhoff (1980).
[KN] Kolmanovskii, V.B. and Nosov, V.R., Stability of Functional Differ-ential Equations, Academic Press, London, Orlando (1986) .
[K] Kushner, H.J., On the stability of processes defined by sto-chastic differential-difference equations, J. Differential equations, 4,(1968), 424-443.
[Ma] Mao, X.R., Exponential Stability of Stochastic Differential Equations, Pureand Applied Mathematics, Marcel Dekker, New York-Basel-Hong Kong (1994).
[MT] Mizel, V.J. and Trutzer, V., Stochastic hereditary equations:existence and asymptotic stability, J. Integral Equations, (1984),1-72
10
[M1] Mohammed, S.-E.A., Stochastic Functional Differential Equations, Re-search Notes in Mathematics 99, Pitman Advanced PublishingProgram, Boston-London-Melbourne (1984).
[M2] Mohammed, S.-E.A., Non-linear flows for linear stochastic delayequations, Stochastics 17, 3 (1986), 207–212.
[M3] Mohammed, S.-E.A., The Lyapunov spectrum and stable mani-folds for stochastic linear delay equations, Stochastics and StochasticReports 29 (1990), 89–131.
[M4] Mohammed, S.-E.A., Lyapunov exponents and stochastic flowsof linear and affine hereditary systems, (1992) (Survey article),in Diffusion Processes and Related Problems in Analysis, Volume II , editedby Pinsky, M., and Wihstutz, V. Birkhauser (1992), 141–169.
[MS] Mohammed, S.-E.A. and Scheutzow, M.K.R., Lyapunov expo-nents of linear stochastic functional differential equation drivenby semimartingales. Part I: The multiplicative ergodic theory,Ann. Inst. Henri Poincare, Probabilites et Statistiques,Vol. 32, no. 1(1996), 69-105.
[PW1] Pardoux, E. and Wihstutz, V., Lyapunov exponent and rota-tion number of two-dimensional stochastic systems with smalldiffusion, SIAM J. Applied Math., 48, (1988), 442-457.
[PW2] Pinsky, M. and Wihstutz, V., Lyapunov exponents of nilpotentIto systems, Stochastics, 25, (1988), 43-57.
[R] Ruelle, D. Characteristic exponents and invariant manifolds inHilbert space, Annals of Mathematics 115 (1982), 243–290.
[S] Scheutzow, M.K.R. Stationary and Periodic Stochastic Differential Sys-tems: A study of Qualitative Changes with Respect to the Noise Level and
11
Asymptotics, Habiltationsschrift, University of Kaiserslautern, W.Germany (1988).
[Sc] Schwartz, L., Radon Measures on Arbitrary Topological Vector Spaces andCylindrical Measures, Tata Institute of Fundamental Research, Ox-ford University, Academic Press, London, Orlando (1986) .
12
I. EXISTENCE
Geilo, Norway
Monday, July 29, 1996
14:00-14:50
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
I. EXISTENCE
1. Examples
Example 1. (Noisy Feedbacks)
Box N: Input = y(t), output = x(t) at time t > 0 related by
x(t) = x(0) +∫ t
0
y(u) dZ(u) (1)
where Z(u) is a semimartingale noise.
Box D: Delays signal x(t) by r (> 0) units of time. A proportion σ
(0 ≤ σ ≤ 1) is transmitted through D and the rest (1 − σ) is used forother purposes.
Thereforey(t) = σx(t− r)
Take Z(u) := white noise = W (u)
2
Then substituting in (1) gives the Ito integral equation
x(t) = x(0) + σ
∫ t
0
x(u− r)dW (u)
or the stochastic differenial delay equation (sdde):
dx(t) = σx(t− r)dW (t), t > 0 (I)
To solve (I), need an initial process θ(t), −r ≤ t ≤ 0:
x(t) = θ(t) a.s., − r ≤ t ≤ 0
r = 0: (I) becomes a linear stochastic ode and has closed form solution
x(t) = x(0)eσW (t)−σ2t2 , t ≥ 0.
r>0: Solve (I) by successive Ito integrations over steps of length r:
x(t) = θ(0) + σ
∫ t
0
θ(u− r) dW (u), 0 ≤ t ≤ r
x(t) = x(r) + σ
∫ t
r
[θ(0) + σ
∫ (v−r)
0
θ(u− r) dW (u)] dW (v), r < t ≤ 2r,
· · · = · · · 2r < t ≤ 3r,
No closed form solution is known (even in deterministic case).Curious Fact!
In the sdde (I) the Ito differential dW may be replaced by theStratonovich differential dW without changing the solution x. Let x
be the solution of (I) under an Ito differential dW . Then using finitepartitions uk of the interval [0, t] :
∫ t
0
x(u− r) dW (t) = lim∑
k
12[x(uk − r) + x(uk+1 − r)][W (uk+1)−W (uk)]
3
where the limit in probability is taken as the mesh of the partitionuk goes to zero. Compare the Stratonovich and Ito integrals usingthe corresponding partial sums:
limE
( ∑
k
12[x(uk − r) + x(uk+1 − r)][W (uk+1)−W (uk)]
−∑
k
[x(uk − r)][W (uk+1)−W (uk)])2
= lim E
(∑
k
12[x(uk+1 − r)− x(uk − r)][W (uk+1)−W (uk)]
)2
= lim∑
k
14E[x(uk+1 − r)− x(uk − r)]2 E[W (uk+1)−W (uk)]2
= lim∑
k
14E[x(uk+1 − r)− x(uk − r)]2 (uk+1 − uk)
= 0
because W has independent increments, x is adapted to the Brownianfiltration, u 7→ x(u) ∈ L2(Ω,R) is continuous, and the delay r is positive.Alternatively
∫ t
0
x(u− r) dW (u) =∫ t
0
x(u− r) dW (u) +12
< x(· − r,W > (t)
and < x(· − r,W > (t) = 0 for all t > 0.
Remark.
When r > 0, the solution process x(t) : t ≥ −r of (I) is a mar-tingale but is non-Markov .Example 2. (Simple Population Growth)
Consider a large population x(t) at time t evolving with a con-stant birth rate β > 0 and a constant death rate α per capita. Assumeimmediate removal of the dead from the population. Let r > 0 (fixed,
4
non-random= 9, e.g.) be the development period of each individualand assume there is migration whose overall rate is distributed likewhite noise σW (mean zero and variance σ > 0), where W is one-dimensional standard Brownian motion. The change in population∆x(t) over a small time interval (t, t + ∆t) is
∆x(t) = −αx(t)∆t + βx(t− r)∆t + σW∆t
Letting ∆t → 0 and using Ito stochastic differentials,
dx(t) = −αx(t) + βx(t− r) dt + σdW (t), t > 0. (II)
Associate with the above affine sdde the initial condition (v, η) ∈ R ×L2([−r, 0],R)
x(0) = v, x(s) = η(s), −r ≤ s < 0.
Denote by M2 = R × L2([−r, 0],R) the Delfour-Mitter Hilbert space ofall pairs (v, η), v ∈ R, η ∈ L2([−r, 0],R) with norm
‖(v, η)‖M2 =(|v|2 +
∫ 0
−r
|η(s)|2 ds
)1/2
.
Let W : R+ × Ω → R be defined on the canonical filtered proba-bility space (Ω,F , (Ft)t∈R+ , P ) where
Ω = C(R+,R), F = Borel Ω, Ft = σρu : u ≤ t
ρu : Ω → R, u ∈ R+, are evaluation maps ω 7→ ω(u), and P = Wienermeasure on Ω.Example 3. (Logistic Population Growth)
A single population x(t) at time t evolving logistically with de-velopment (incubation) period r > 0 under Gaussian type noise (e.g.migration on a molecular level):
x(t) = [α− βx(t− r)] x(t) + γx(t)W (t), t > 0
5
i.e.dx(t) = [α− βx(t− r)] x(t) dt + γx(t)dW (t) t > 0. (III)
with initial condition
x(t) = θ(t) − r ≤ t ≤ 0.
For positive delay r the above sdde can be solved implicitly usingforward steps of length r, i.e. for 0 ≤ t ≤ r, x(t) satisfies the linear sode(without delay)
dx(t) = [α− βθ(t− r)] x(t) dt + γx(t)dW (t) 0 < t ≤ r. (III ′)
x(t) is a semimartingale and is non-Markov (Scheutzow [S], 1984).Example 4. (Heat bath)
Model proposed by R. Kubo (1966) for physical Brownian mo-tion. A molecule of mass m moving under random gas forces withposition ξ(t) and velocity v(t) at time t; cf classical work by Einsteinand Ornestein and Uhlenbeck. Kubo proposed the following modifi-cation of the Ornstein-Uhenbeck process
dξ(t) = v(t) dt
mdv(t) = −m[∫ t
t0
β(t− t′)v(t′) dt′] dt + γ(ξ(t), v(t)) dW (t), t > t0.
(IV )
m =mass of molecule. No external forces.β = viscosity coefficient function with compact support.γ a function R3 ×R3 → R representing the random gas forces on
the molecule.ξ(t) = position of molecule ∈ R3.v(t) = velocity of molecule ∈ R3.W = 3− dimensional Brownian motion.
([Mo], Pitman Books, RN # 99, 1984, pp. 223-226).
6
Further Examples
Delay equation with Poisson noise:
dx(t) = x((t− r)−) dN(t) t > 0
x0 = η ∈ D([−r, 0],R)
(V )
N := Poisson process with iid interarrival times ([S], Hab. 1988).D([−r, 0],R) = space of all cadlag paths [−r, 0] → R, with sup norm.
Simple model of dye circulation in the blood (or pollution) (cf.Bailey and Williams [B-W], JMAA, 1966, Lenhart and Travis ([L-T],PAMS, 1986).
dx(t) = νx(t) + µx(t− r)) dt + σx(t) dW (t) t > 0
(x(0), x0) = (v, η) ∈ M2 = R× L2([−r, 0],R),
(V I)
([Mo], Survey, 1992; [M-S], II, 1995.)In above model:x(t) := dye concentration (gm/cc)r = time taken by blood to traverse side tube (vessel)Flow rate (cc/sec) is Gaussian with variance σ.A fixed proportion of blood in main vessel is pumped into side
vessel(s). Model will be analysed in Lecture V (Theorem V.5).
7
dx(t) = νx(t) + µx(t− r)) dt + ∫ 0
−r
x(t + s)σ(s) ds dW (t),
(x(0), x0) = (v, η) ∈ M2 = R× L2([−r, 0],R), t > 0.
(V II)
([Mo], Survey, 1992; [M-S], II, 1995.)
Linear d-dimensional systems driven by m-dimensional Brown-ian motion W := (W1, · · · , Wm) with constant coefficients.
dx(t) = H(x(t− d1), · · · , x(t− dN ), x(t), xt)dt
+m∑
i=1
gix(t) dWi(t), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(V III)
H := (Rd)N ×M2 → Rd linear functional on (Rd)N ×M2; gi d× d-matrices([Mo], Stochastics, 1990).
Linear systems driven by (helix) semimartingale noise (N, L),and memory driven by a (stationary) measure-valued process ν and a(stationary) process K ([M-S], I, AIHP, 1996):
dx(t) =∫
[−r,0]
ν(t)(ds)x(t + s)
dt
+ dN(t)∫ 0
−r
K(t)(s)x(t + s) ds + dL(t) x(t−), t > 0
(x(0), x0) = (v, η) ∈ M2 = Rd × L2([−r, 0],Rd)
(IX)
Multidimensional affine systems driven by (helix) noise Q ([M-S], Stochastics, 1990):
dx(t) =∫
[−r,0]
ν(t)(ds)x(t + s)
dt + dQ(t), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(X)
8
Memory driven by white noise:
dx(t) =∫
[−r,0]
x(t + s) dW (s)
dW (t) t > 0
x(0) = v ∈ R, x(s) = η(s), −r < s < 0, r ≥ 0
(XI)
([Mo], Survey, 1992).
9
Formulation
Slice each solution path x over the interval [t − r, t] to get segment xt
as a process on [−r, 0]:
xt(s) := x(t + s) a.s., t ≥ 0, s ∈ J := [−r, 0].
Therefore sdde’s (I), (II), (III) and (XI) become
dx(t) = σxt(−r)dW (t), t > 0
x0 = θ ∈ C([−r, 0],R)
(I)
dx(t) = −αx(t) + βxt(−r) dt + σdW (t), t > 0
(x(0), x0) = (v, η) ∈ R× L2([−r, 0],R)
(II)
10
dx(t) = [α− βxt(−r)]xt(0) dt + γxt(0) dW (t)
x0 = θ ∈ C([−r, 0],R)
(III)
dx(t) =∫
[−r,0]
xt(s) dW (s)
dW (t) t > 0
(x(0), x0) = (v, η) ∈ R× L2([−r, 0],R), r ≥ 0
(XI)
Think of R.H.S.’s of the above equations as functionals of xt
(and x(t)) and generalize to stochastic functional differential equation
(sfde)
dx(t) = h(t, xt)dt + g(t, xt)dW (t) t > 0
x0 = θ
(XII)
on filtered probability space (Ω,F , (Ft)t≥0, P ) satisfying the usual con-
ditions:
(Ft)t≥0 right-continuous and each Ft contains all P -null sets in
F.
C := C([−r, 0],Rd) Banach space, sup norm.
W (t) = m–dimensional Brownian motion.
11
L2(Ω, C) := Banach space of all (F , BorelC)-measurable L2 (Bochner
sense) maps Ω → C with the L2-norm
‖θ‖L2(Ω,C) :=[∫
Ω
‖θ(ω)‖2C dP (ω)]1/2
Coefficients:
h : [0, T ]× L2(Ω, C) → L2(Ω,Rd) (Drift)
g : [0, T ]× L2(Ω, C) → L2(Ω, L(Rm,Rd) (Diffusion).
Initial data:
θ ∈ L2(Ω, C,F0).
Solution:
x : [−r, T ]×Ω → Rd measurable and sample-continuous, x|[0, T ] (Ft)0≤t≤T -
adapted and x(s) is F0-measurable for all s ∈ [−r, 0].
Exercise: [0, T ] 3 t 7→ xt ∈ C([−r, 0],Rd) is (Ft)0≤t≤T -adapted.
(Hint: Borel C is generated by all evaluations.)
12
Hypotheses (E1).
(i) h, g are jointly continuous and uniformly Lipschitz in the second
variable with respect to the first:
‖h(t, ψ1)− h(t, ψ2)‖L2(Ω,Rd) ≤ L‖ψ1 − ψ2‖L2(Ω,C)
for all t ∈ [0, T ] and ψ1, ψ2 ∈ L2(Ω, C). Similarly for the diffusion
coefficent g.
(ii) For each (Ft)0≤t≤T -adapted process y : [0, T ] → L2(Ω, C),
the processes h(·, y(·)), g(·, y(·)) are also (Ft)0≤t≤T - adapted.
Theorem I.1. ([Mo], 1984) (Existence and Uniqueness).
Suppose h and g satisfy Hypotheses (E1). Let θ ∈ L2(Ω, C;F0).
Then the sfde (XII) has a unique solution θx : [−r,∞) × Ω → Rd starting
off at θ ∈ L2(Ω, C;F0) with t 7−→ θxt continuous and θx ∈ L2(Ω, C([−r, T ]Rd)) for
all T > 0. For a given θ, uniqueness holds up to equivalence among all (Ft)0≤t≤T -
adapted processes in L2(Ω, C([−r, T ],Rd)).
Proof.
[Mo], Pitman Books, 1984, Theorem 2.1, pp. 36-39. ¤
13
Theorem I.1 covers equations (I), (II), (IV), (VI), (VII), (VIII),
(XI) and a large class of sfde’s driven by white noise. Note that
(XI) does not satisfy the hypotheses underlying the classical results
of Doleans-Dade [Dol], 1976, Metivier and Pellaumail [Met-P], 1980,
Protter, Ann. Prob. 1987, Lipster and Shiryayev [Lip-Sh], [Met],
1982. This is because the coefficient
η →∫ 0
−r
η(s) dW (s)
on the RHS of (XI) does not admit almost surely Lipschitz (or even
linear) versions C → R! This will be shown later.
When the coeffcients h, g factor through functionals
H : [0, T ]× C → Rd, G : [0, T ]× C → Rd×m
we can impose the following local Lipschitz and global linear growth
conditions on the sfde
dx(t) = H(t, xt) dt + G(t, xt) dW (t) t > 0
x0 = θ
(XIII)
with W m-dimensional Brownian motion:
Hypotheses (E2)
14
(i) H, G are Lipschitz on bounded sets in C: For each integer n ≥ 1
there exists Ln > 0 such that
|H(t, η1)−H(t, η2)| ≤ Ln‖η1 − η2‖C
for all t ∈ [0, T ] and η1, η2 ∈ C with ‖η1‖C ≤ n, ‖η2‖C ≤ n. Similarly
for the diffusion coefficent G.
(ii) There is a constant K > 0 such that
|H(t, η)|+ ‖G(t, η)‖ ≤ K(1 + ‖η‖C)
for all t ∈ [0, T ] and η ∈ C.
Note that the adaptability condition is not needed (explicitly)
because H, G are deterministic and because the sample-continuity and
adaptability of x imply that the segment [0, T ] 3 t 7→ xt ∈ C is also
adapted.
Exercise: Formulate the heat-bath model (IV) as a sfde of the form
(XIII).(β has compact support in R+.)
Theorem I.2. ([Mo], 1984) (Existence and Uniqueness).
Suppose H and G satisfy Hypotheses (E2) and let θ ∈ L2(Ω, C;F0).
15
Then the sfde (XIII) has a unique (Ft)0≤t≤T -adapted solution θx : [−r, T ]×
Ω → Rd starting off at θ ∈ L2(Ω, C;F0) with t 7−→ θxt continuous and θx ∈
L2(Ω, C([−r, T ],Rd)) for all T > 0. For a given θ, uniqueness holds up to equiva-
lence among all (Ft)0≤t≤T -adapted processes in L2(Ω, C([−r, T ],Rd)).
Furthermore if θ ∈ L2k(Ω, C;F0), then θxt ∈ L2k(Ω, C;Ft) and
E‖θxt‖2kC ≤ Ck[1 + ‖θ‖2k
L2k(Ω,C)]
for all t ∈ [0, T ] and some positive constants Ck.
16
Proofs of Theorems I.1, I.2.(Outline)
[Mo], pp. 150-152. Generalize sode proofs in Gihman and Sko-
rohod ([G-S], 1973) or Friedman ([Fr], 1975):
(1) Truncate coefficients outside bounded sets in C. Reduce to glob-
ally Lipschitz case.
(2) Successive approx. in globally Lipschitz situation.
(3) Use local uniqueness ([Mo], Theorem 4.2, p. 151) to “patch up”
solutions of the truncated sfde’s.
For (2) consider globally Lipschitz case and h ≡ 0.
We look for solutions of (XII) by successive approximation in
L2(Ω, C([−r, a],Rd)). Let J := [−r, 0].
Suppose θ ∈ L2(Ω, C(J,Rd)) is F0-measurable. Note that this is
equivalent to saying that θ(·)(s) is F0-measurable for all s ∈ J, because
θ has a.a. sample paths continuous.
We prove by induction that there is a sequence of processes
kx : [−r, a]× Ω → Rd, k = 1, 2, · · · having the
Properties P (k):
17
(i) kx ∈ L2(Ω, C([−r, a],Rd)) and is adapted to (Ft)t∈[0,a].
(ii) For each t ∈ [0, a], kxt ∈ L2(Ω, C(J,Rd)) and is Ft-measur-able.
(iii)
‖k+1x− kx‖L2(Ω,C) ≤ (ML2)k−1 ak−1
(k − 1)!‖2x− 1x‖L2(Ω,C)
‖k+1xt − kxt‖L2(Ω,C) ≤ (ML2)k−1 tk−1
(k − 1)!‖2x− 1x‖L2(Ω,C)
(1)
where M is a “martingale” constant and L is the Lipschitz constant
of g.
Take 1x : [−r, a]× Ω → Rd to be
1x(t, ω) =
θ(ω)(0) t ∈ [0, a]θ(ω)(t) t ∈ J
a.s., and
k+1x(t, ω) =
θ(ω)(0) + (ω)∫ t
0
g(u, kxu)dW (·)(u) t ∈ [0, a]
θ(ω)(t) t ∈ J
(2)
a.s.
Since θ ∈ L2(Ω, C(J,Rd)) and is F0-measurable, then 1x ∈ L2(Ω, C([−r, a],Rd))
and is trivially adapted to (Ft)t∈[0,a]. Hence 1xt ∈ L2(Ω, C(J,Rd)) and is
Ft-measurable for all t ∈ [0, a]. P (1) (iii) holds trivially.
18
Now suppose P (k) is satisfied for some k > 1. Then by Hypothesis
(E1)(i), (ii) and the continuity of the slicing map (stochastic memory),
it follows from P (k)(ii) that the process
[0, a] 3 u 7−→ g(u, kxu) ∈ L2(Ω, L(Rm,Rd))
is continuous and adapted to (Ft)t∈[0,a]. P (k+1)(i) and P (k+1)(ii) follow
from the continuity and adaptability of the stochastic integral. Check
P (k + 1)(iii), by using Doob’s inequality.
For each k > 1, write
kx = 1x +k−1∑
i=1
(i+1x− ix).
Now L2A(Ω, C([−r, a],Rd)) is closed in L2(Ω, C([−r, a],Rd)); so the series
∞∑
i=1
(i+1x− ix)
converges in L2A(Ω, C([−r, a],Rd)) because of (1) and the convergence of
∞∑
i=1
[(ML2)i−1 ai−1
(i− 1)!
]1/2
.
Hence kx∞k=1 converges to some x ∈ L2A(Ω, C([−r, a],Rd)).
19
Clearly x|J = θ and is F0-measurable, so applying Doob’s in-
equality to the Ito integral of the difference
u 7−→ g(u, kxu)− g(u, xu)
gives
E
(sup
t∈[0,a]
∣∣∣∣∫ t
0
g(u, kxu) dW (·)(u)−∫ t
0
g(u, xu) dW (·)(u)∣∣∣∣2)
< ML2a‖kx− x‖2L2(Ω,C)
−→ 0 as k →∞.
Thus viewing the right-hand side of (2) as a process in L2(Ω, C ([−r, a],Rd))
and letting k → ∞, it follows from the above that x must satisfy the
sfde (XII) a.s. for all t ∈ [−r, a].
For uniqueness, let x ∈ L2A(Ω, ([−r, a],Rd)) be also a solution of
(XII) with initial process θ. Then by the Lipschitz condition:
‖xt − xt‖2L2(Ω,C) < ML2
∫ t
0
‖xu − xu‖2L2(Ω,C) du
for all t ∈ [0, a]. Therefore we must have xt − xt = 0 for all t ∈ [0, a]; so
x = x in L2(Ω, C([−r, a],Rd)) a.s. ¤
20
Remarks and Generalizations.
(i) In Theorem I.2 replace the process (t,W (t)) by a (square inte-
grable) semimartingale Z(t) satisfying appropriate conditions.([Mo],
1984, Chapter II).
(ii) Results on existence of solutions of sfde’s driven by white noise
were first obtained by Ito and Nisio ([I-N], J. Math. Kyoto
University, 1968) and then Kushner (JDE, 197).
(iii) Extensions to sfde’s with infinite memory. Fading memory case:
work by Mizel and Trutzer [M-T],JIE, 1984, Marcus and Mizel
[M-M], Stochastics, 1988; general infinite memory: Ito and Nisio
[I-N], J. Math. Kyoto University, 1968.
(iii) Pathwise local uniqueness holds for sfde’s of type (XIII) under
a global Lipschitz condition: If coeffcients of two sfde’s agree
on an open set in C, then the corresponding trajectories leave
the open set at the same time and agree almost surely up to
the time they leave the open set ([Mo], Pitman Books, 1984,
Theorem 4.2, pp. 150-151.)
21
(iv) Replace the state space C by the Delfour-Mitter Hilbert space
M2 := Rd × L2([−r, 0],Rd) with the Hilbert norm
‖(v, η)‖M2 =(|v|2 +
∫ 0
−r
|η(s)|2 ds
)1/2
for (v, η) ∈ M2 (T. Ahmed, S. Elsanousi and S. Mohammed,
1983).
(v) Have Lipschitz and smooth dependence of θxt on the initial pro-
cess θ ∈ L2(Ω, C) ([Mo], 1984, Theorems 3.1, 3.2, pp. 41-45).
22
II. MARKOV BEHAVIOR
AND THE WEAK GENERATOR
Geilo, Norway
Tuesday, July 30, 1996
14:00-14:50
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
II. MARKOV BEHAVIOR AND THE GENERATOR
Consider the sfde
dx(t) = H(t, xt) dt + G(t, xt) dW (t), t > 0
x0 = η ∈ C := C([−r, 0],Rd)
(XIII)
with coefficients H : [0, T ]×C → Rd, G : [0, T ]×C → Rd×m, m-dimensionalBrownian motion W and trajectory field ηxt : t ≥ 0, η ∈ C.
1. Questions
(i) For the sfde (XIII) does the trajectory field xt give a diffusionin C (or M2)?
(ii) How does the trajectory xt transform under smooth non-linearfunctionals φ : C → R?
(iii) What “diffusions” on C (or M2) correspond to sfde’s on Rd?
We will only answer the first two questions. More details in[Mo], Pitman Books, 1984, Chapter III, pp. 46-112. Third questionis OPEN.
2
Difficulties
(i) Although the current state x(t) is a semimartingale, the trajec-tory xt does not seem to possess any martingale properties whenviewed as C-(or M2)-valued process: e.g. for Brownian motionW (H ≡ 0, G ≡ 1):
[E(Wt|Ft1)](s) = W (t1) = Wt1(0), s ∈ [−r, 0]
whenever t1 ≤ t− r.(ii) Lack of strong continuity leads to the use of weak limits in C
which tend to live outside C.(iii) We will show that xt is a Markov process in C. However al-
most all tame functions lie outside the domain of the (weak)generator.
(iv) Lack of an Ito formula makes the computation of the generatorhard.
Hypotheses (M)
(i) Ft := completion of σW (u) : 0 ≤ u ≤ t, t ≥ 0.(ii) H, G are jointly continuous and globally Lipschitz in second vari-
able uniformly wrt the first:
|H(t, η1)−H(t, η2)|+ ‖G(t, η1)−G(t, η2)‖ ≤ L‖η1 − η2‖C
for all t ∈ [0, T ] and η1, η2 ∈ C.
2. The Markov Propertyηxt1 := solution starting off at θ ∈ L2(Ω, C;Ft1) at t = t1 for the
sfde:
ηxt1(t) =
η(0) +
∫ t
t1H(u, xt1
u ) du +∫ t
t1G(u, xt1
u ) dW (u), t > t1
η(t− t1), t1 − r ≤ t ≤ t1.
3
This gives a two-parameter family of mappings
T t1t2 : L2(Ω, C;Ft1) → L2(Ω, C;Ft2), t1 ≤ t2,
T t1t2 (θ) := θxt1
t2 , θ ∈ L2(Ω, C;Ft1). (1)
Uniqueness of solutions gives the two-parameter semigroup property:
T t1t2 T 0
t1 = T 0t2 , t1 ≤ t2. (2)
([Mo], Pitman Books, 1984, Theorem II (2.2), p. 40.)
Theorem II.1 (Markov Property)([Mo], 1984).In (XIII) suppose Hypotheses (M) hold. Then the trajectory field ηxt : t ≥
0, η ∈ C is a Feller process on C with transition probabilities
p(t1, η, t2, B) := P(ηxt1
t2 ∈ B)
t1 ≤ t2, B ∈ Borel C, η ∈ C.
i.e.
P(xt2 ∈ B
∣∣Ft1
)= p(t1, xt1(·), t2, B) = P
(xt2 ∈ B
∣∣xt1
)a.s.
Further, if H and G do not depend on t, then the trajectory is time-homogeneous:
p(t1, η, t2, ·) = p(0, η, t2 − t1, ·), 0 ≤ t1 ≤ t2, η ∈ C.
Proof.
[Mo], 1984, Theorem III.1.1, pp. 51-58. [Mo], 1984, TheoremIII.2.1, pp. 64-65. ¤
4
3. The Semigroup
In the autonomous sfde
dx(t) = H(xt) dt + G(xt) dW (t) t > 0
x0 = η ∈ C
(XIV )
suppose the coefficients H : C → Rd, G : C → Rd×m are globallybounded and globally Lipschitz.Cb := Banach space of all bounded uniformly continuous functionsφ : C → R, with the sup norm
‖φ‖Cb:= sup
η∈C|φ(η)|, φ ∈ Cb.
Define the operators Pt : Cb → Cb, t ≥ 0, on Cb by
Pt(φ)(η) := Eφ(ηxt
)t ≥ 0, η ∈ C.
A family φt, t > 0, converges weakly to φ ∈ Cb as t → 0+ if limt→0+
<
φt, µ >=< φ, µ > for all finite regular Borel measures µ on C. Writeφ := w − lim
t→0+φt. This is equivalent to
φt(η) → φ(η) as t → 0+, for all η ∈ C
‖φt‖Cb: t ≥ 0 is bounded .
(Dynkin, [Dy], Vol. 1, p. 50). Proof uses uniform boundednessprinciple and dominated convergence theorem.
Theorem II.2([Mo], Pitman Books, 1984)(i) Ptt≥0 is a one-parameter contraction semigroup on Cb.
5
(ii) Ptt≥0 is weakly continuous at t = 0:
Pt(φ)(η) → φ(η) as t → 0+
|Pt(φ)(η)| : t ≥ 0, η ∈ Cis bounded by ‖φ‖Cb.
(iii) If r > 0, Ptt≥0 is never strongly continuous on Cb under the sup norm.
Proof.
(i) One parameter semigroup property
Pt2 Pt1 = Pt1+t2 , t1, t2 ≥ 0
follows from the continuation property (2) and time-homogeneityof the Feller process xt (Theorem II.1).
(ii) Definition of Pt, continuity and boundedness of φ and sample-continuity of trajectory ηxt give weak continuity of Pt(φ) : t > 0at t = 0 in Cb.
(iii) Lack of strong continuity of semigroup:Define the canonical shift (static) semigroup
St : Cb → Cb, t ≥ 0,
bySt(φ)(η) := φ(ηt), φ ∈ Cb, η ∈ C,
where η : [−r,∞) → Rd is defined by
η(t) =
η(0) t ≥ 0η(t) t ∈ [−r, 0).
Then Pt is strongly continuous iff St is strongly continuous. Pt
and St have the same “domain of strong continuity” indepen-dently of H, G, and W . This follows from the global bound-edness of H and G. ([Mo], Theorem IV.2.1, pp. 72-73). Keyrelation is
limt→0+
E‖ηxt − ηt‖2C = 0
6
uniformly in η ∈ C. But St is strongly continuous on Cb iff C islocally compact iff r = 0 (no memory) ! ([Mo], Theorems IV.2.1and IV.2.2, pp.72-73). Main idea is to pick any s0 ∈ [−r, 0) andconsider the function φ0 : C → R defined by
φ0(η) :=
η(s0) ‖η‖C ≤ 1η(s0)‖η‖C
‖η‖C > 1
Let C0b be the domain of strong continuity of Pt, viz.
C0b := φ ∈ Cb : Pt(φ) → φ as t → 0+ in Cb.
Then φ0 ∈ Cb, but φ0 /∈ C0b because r > 0. ¤
4. The Generator
Define the weak generator A : D(A) ⊂ Cb → Cb by the weak limit
A(φ)(η) := w − limt→0+
Pt(φ)(η)− φ(η)t
where φ ∈ D(A) iff the above weak limit exists. Hence D(A) ⊂ Cb0
(Dynkin [Dy], Vol. 1, Chapter I, pp. 36-43). Also D(A) is weaklydense in Cb and A is weakly closed. Further
d
dtPt(φ) = A(Pt(φ)) = Pt(A(φ)), t > 0
for all φ ∈ D(A) ([Dy], pp. 36-43).
Next objective is to derive a formula for the weak generatorA. We need to augment C by adjoining a canonical d-dimensionaldirection. The generator A will be equal to the weak generator ofthe shift semigroup St plus a second order linear partial differentialoperator along this new direction. Computation requires the followinglemmas.
Let
Fd = vχ0 : v ∈ RdC ⊕ Fd = η + vχ0 : η ∈ C, v ∈ Rd, ‖η + vχ0‖ = ‖η‖C + |v|
7
Lemma II.1.([Mo], Pitman Books, 1984)Suppose φ : C → R is C2 and η ∈ C. Then Dφ(η) and D2φ(η) have unique
weakly continuous linear and bilinear extensions
Dφ(η) : C ⊕ Fd → R, D2φ(η) : (C ⊕ Fd)× (C ⊕ Fd) → R
respectively.
Proof.
First reduce to the one-dimensional case d = 1 by using coordi-nates.
Let α ∈ C∗ = [C([−r, 0],R)]∗. We will show that there is a weaklycontinuous linear extension α : C⊕F1 → R of α; viz. If ξk is a boundedsequence in C such that ξk(s) → ξ(s) as k → ∞ for all s ∈ [−r, 0], whereξ ∈ C ⊕ F1, then α(ξk) → α(ξ) as k → ∞. By the Riesz representationtheorem there is a unique finite regular Borel measure µ on [−r, 0] suchthat
α(η) =∫ 0
−r
η(s) dµ(s)
for all η ∈ C. Define α ∈ [C ⊕ F1]∗ by
α(η + vχ0) = α(η) + vµ(0), η ∈ C, v ∈ R.
Easy to check that α is weakly continuous. (Exercise: Use Lebesguedominated convergence theorem.)
Weak extension α is unique because each function vχ0 can beapproximated weakly by a sequence of continuous functions ξk
0:
ξk0 (s) :=
(ks + 1)v, − 1
k ≤ s ≤ 0
0 − r ≤ s < − 1k .
8
Put α = Dφ(η) to get first assertion of lemma.To construct a weakly continuous bilinear extension β : (C⊕F1)×
(C ⊕ F1) → R for any continuous bilinear formβ : C × C → R, use classical theory of vector measures (Dunford andSchwartz, [D-S], Vol. I, Section 6.3). Think of β as a continuos linearmap C → C∗. Since C∗ is weakly complete ([D-S], I.13.22, p. 341),then β is a weakly compact linear operator ([D-S], Theorem I.7.6, p.494): i.e. it maps norm-bounded sets in C into weakly sequentiallycompact sets in C∗. By the Riesz representation theorem (for vectormeasures), there is a unique C∗-valued Borel measure λ on [−r, 0] (offinite semi-variation) such that
β(ξ) =∫ 0
−r
ξ(s) dλ(s)
for all ξ ∈ C. ([D-S], Vol. I, Theorem VI.7.3, p. 493). By thedominated convergence theorem for vector measures ([D-S], Theo-rem IV.10.10, p. 328), one could reach elements in F1 using weaklyconvergent sequences of type ξk
0. This gives a unique weakly con-tinuous extension β : C ⊕ F1 → C∗. Next for each η ∈ C, v ∈ R, extendβ(η + vχ0) : C → R to a weakly continuous linear map β(η + vχ0) :C ⊕ F1 → R. Thus β corresponds to the weakly continuous bilinearextension β(·)(·) : [C ⊕ F1]× [C ⊕ F1] → R of β. (Check this as exercise).
9
Finally use β = D2φ(η) for each fixed η ∈ C to get the requiredbilinear extension D2φ(η). ¤
Lemma II.2. ([Mo], Pitman Books, 1984)For t > 0 define W ∗
t ∈ C by
W ∗t (s) :=
1√t[W (t + s)−W (0)], −t ≤ s < 0,
0 − r ≤ s ≤ −t.
Let β be a continuous bilinear form on C. Then
limt→0+
[1tEβ(ηxt − ηt,
ηxt − ηt)− Eβ(G(η) W ∗t , G(η) W ∗
t )]
= 0
Proof.
Uselim
t→0+E‖ 1√
t(ηxt − ηt −G(η) W ∗
t ‖2C = 0.
The above limit follows from the Lipschitz continuity of H and G andthe martingale properties of the Ito integral. Conclusion of lemmais obtained by a computation using the bilinearity of β, Holder’s in-equality and the above limit.([Mo], Pitman Books, 1984, pp. 86-87.)
¤
Lemma II.3. ([Mo], Pitman Books, 1984)Let β be a continuous bilinear form on C and eim
i=1 be any basis for Rm.
Then
limt→0+
1tEβ(ηxt − ηt,
ηxt − ηt) =m∑
i=1
β(G(η)(ei)χ0, G(η)(ei)χ0
)
for each η ∈ C.
Proof.
10
By taking coordinates reduce to the one-dimensional case d =m = 1:
limt→0+
Eβ(W ∗t ,W ∗
t ) = β(χ0, χ0)
with W one-dimensional Brownian motion. The proof of the aboverelation is lengthy and difficult. A key idea is the use of the projectivetensor product C ⊗π C in order to view the continuous bilinear form β
as a continuous linear functional on C ⊗π C. At this level β commuteswith the (Bochner) expectation. Rest of computation is effected usingMercer’s theorem and some Fourier analysis. See [Mo], 1984, pp. 88-94. ¤Theorem II.3.([Mo], Pitman Books, 1984)
In (XIV) suppose H and G are globally bounded and Lipschitz. Let S :D(S) ⊂ Cb → Cb be the weak generator of St. Suppose φ ∈ D(S) is sufficiently
smooth (e.g. φ is C2, Dφ, D2φ globally bounded and Lipschitz). Then φ ∈ D(A)and
A(φ)(η) = S(φ)(η) + Dφ(η)(H(η)χ0
)
+12
m∑
i=1
D2φ(η)(G(η)(ei)χ0, G(η)(ei)χ0
).
where eimi=1 is any basis for Rm.
Proof.
Step 1.For fixed η ∈ C, use Taylor’s theorem:
φ(ηxt)− φ(η) = φ(ηt)− φ(η) + Dφ(ηt)(ηxt − ηt) + R(t)
a.s. for t > 0; where
R(t) :=∫ 1
0
(1− u)D2φ[ηt + u(ηxt − ηt)](ηxt − ηt,ηxt − ηt) du.
11
Take expectations and divide by t > 0:
1tE[φ(ηxt)− φ(η)] =
1t[St(φ(η)−φ(η)] + Dφ(ηt)
E[
1t(ηxt − ηt)]
+1tER(t)
(3)
for t > 0.As t → 0+, the first term on the RHS converges to S(φ)(η), be-
cause φ ∈ D(S).Step 2.
Consider second term on the RHS of (3). Then
limt→0+
[E
1t(ηxt − ηt)
](s) =
limt→0+
1t
∫ t
0
E[H(ηxu)] du, s = 0
0 − r ≤ s < 0.
= [H(η)χ0](s), −r ≤ s ≤ 0.
Since H is bounded, then ‖E1t (
ηxt − ηt)‖C is bounded in t > 0 and
η ∈ C (Exercise). Hence
w − limt→0+
[E
1t(ηxt − ηt)
]= H(η)χ0 (/∈ C).
Therefore by Lemma II.1 and the continuity of Dφ at η:
limt→0+
Dφ(ηt)
E
[1t(ηxt − ηt)
]= lim
t→0+Dφ(η)
E
[1t(ηxt − ηt)
]
= Dφ(η)(H(η)χ0
)
Step 3.
12
To compute limit of third term in RHS of (3), consider∣∣∣∣1tED2φ[ηt + u(ηxt − ηt)](ηxt − ηt,
ηxt − ηt)
− 1tED2φ(η)(ηxt − ηt,
ηxt − ηt)∣∣∣∣
≤ (E‖D2φ[ηt + u(ηxt − ηt)]−D2φ(η)‖2)1/2
[1t2
E‖ηxt − ηt‖4]1/2
≤ K(t2 + 1)1/2[E‖D2φ[ηt + u(ηxt − ηt)]−D2φ(η)‖2]1/2
→ 0
as t → 0+, uniformly for u ∈ [0, 1], by martingale properties of the Itointegral and the Lipschitz continuity of D2φ. Therefore by Lemma II.3
limt→0+
1tER(t) =
∫ 1
0
(1− u) limt→0+
1tED2φ(η)(ηxt − ηt,
ηxt − ηt) du
=12
m∑
i=1
D2φ(η)(G(η)(ei)χ0, G(η)(ei)χ0
).
The above is a weak limit since φ ∈ D(S) and has first and secondderivatives globally bounded on C. ¤
5. Quasitame Functions
Recall that a function φ : C → R is tame (or a cylinder function)if there is a finite set s1 < s2 < · · · < sk in [−r, 0] and a C∞-boundedfunction f : (Rd)k → R such that
φ(η) = f(η(s1), · · · , η(sk)), η ∈ C.
The set of all tame functions is a weakly dense subalgebra ofCb, invariant under the static shift St and generates Borel C. For k ≥ 2the tame function φ lies outside the domain of strong continuity C0
b ofPt, and hence outside D(A) ([Mo], Pitman Books, 1984, pp.98-103; seealso proof of Theorem IV .2.2, pp. 73-76). To overcome this difficultywe introduce
13
Definition.
Say φ : C → R is quasitame if there are C∞-bounded maps h :(Rd)k → R, fj : Rd → Rd, and piecewise C1 functions gj : [−r, 0] → R, 1 ≤j ≤ k − 1, such that
φ(η) = h
(∫ 0
−r
f1(η(s))g1(s) ds, · · · ,
∫ 0
−r
fk−1(η(s))gk−1(s) ds, η(0))
(4)
for all η ∈ C.Theorem II.4. ([Mo], Pitman Books, 1984)
The set of all quasitame functions is a weakly dense subalgebra of C0b , in-
variant under St, generates Borel C and belongs to D(A). In particular, if φ is the
quasitame function given by (4), then
A(φ)(η) =k−1∑
j=1
Djh(m(η))fj(η(0))gj(0)− fj(η(−r))gj(−r)
−∫ 0
−r
fj(η(s))g′j(s) ds
+ Dkh(m(η))(H(η)) +12trace[D2
kh(m(η)) (G(η)×G(η))].
for all η ∈ C, where
m(η) :=(∫ 0
−r
f1(η(s))g1(s) ds, · · · ,
∫ 0
−r
fk−1(η(s))gk−1(s) ds, η(0))
.
Remarks.
(i) Replace C by the Hilbert space M2. No need for the weak ex-tensions because M2 is weakly complete. Extensions of Dφ(v, η)and D2φ(v, η) correspond to partial derivatives in the Rd-variable.Tame functions do not exist on M2 but quasitame functions do!(with η(0) replaced by v ∈ Rd).
14
Analysis of supermartingale behavior and stability of φ(ηxt) givenin Kushner ([Ku], JDE, 1968). Infinite fading memory settingby Mizel and Trutzer ([M-T], JIE, 1984) in the weighted statespace Rd × L2((−∞, 0],Rd; ρ).
(ii) For each quasitame φ on C, φ(ηxt) is a semimartingale, and theIto formula holds:
d[φ(ηxt)] = A(φ)(ηxt) dt + Dφ(η)(H(η)χ0
)dW (t).
15
III. REGULARITY
CLASSIFICATION OF SFDE’S
Geilo, Norway
Wednesday, July 31, 1996
10:30-11:20
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
III. REGULARITY. CLASSIFICATION OF SFDE’S
Denote the state space by E where E = C or M2 := Rd×L2([−r, 0],Rd).Most results hold for either choice of state space.Objectives
To study regularity properties of the trajectory of a sfde as arandom field X := ηxt : t ≥ 0, η ∈ C in the variables (t, η, ω) (E = C) or(t, (v, η), ω) (E = M2):
(i) Pathwise regularity of trajectories in the time variable.(ii) Regularity of trajectories (in probability or pathwise) in the
initial state η ∈ C or (v, η) ∈ M2.(iii) Classification of sfde’s into regular and singular types.
Denote by Cα := Cα([−r, 0],Rd) the (separable) Banach space ofα-Holder continuous paths η : [−r, 0] → Rd with the Holder norm
‖η‖α := ‖η‖C + sup |η(s1)− η(s2)|
|s1 − s2|α : s1, s2 ∈ [−r, 0], s1 6= s2
.
Cα can be constructed in a separable manner by completing the spaceof smooth paths [−r, 0] → Rd with respect to the above norm (Tromba[Tr], JFA, 1972). First step is to think of ηxt(ω) as a measurablemapping X : R+ × C × Ω → C in the three variables (t, η, ω) simultane-ously:
Theorem III.1([Mo], Pitman Books, 1984)In the sfde
dx(t) = H(t, xt) dt + G(t, xt) dW (t) t > 0
x0 = η ∈ C
(XIII)
assume that the coefficients H, G are (jointly) continuous and globally Lipschitz in
the second variable uniformly wrt the first. Then2
(i) For any 0 < α <12, and each initial path η ∈ C,
P (ηxt ∈ Cα, for all t ≥ r) = 1.
(ii) the trajectory field has a measurable version
X : R+ × C × Ω → C.
(iii) The trajectory field ηxt, t ≥ r, η ∈ C, admits a measurable version
[r,∞)× C × Ω → Cα.
Remark.
Similar statements hold for E = M2.Give L0(Ω, E) the complete (psuedo)metric
dE(θ1, θ2) := infε>0
[ε + P (‖θ1 − θ2‖E ≥ ε)], θ1, θ2 ∈ L0(Ω, E),
(which corresponds to convergence in probability, Dunford and Schwartz[D-S], Lemma III.2.7, p. 104).Proof of Theorem III.1.
(i) Sufficient to show that
P(ηx|[0, a] ∈ Cα([0, a],Rd)
)= 1
by using the estimate
P
(sup
0≤t1,t2≤a,t1 6=t1
|ηx(t1)− ηx(t2)||t1 − t2|α ≥ N
)≤ C1
k(1 + ‖η‖2kC )
1N2k
,
for all integers k > (1 − 2α)−1, and the Borel-Cantelli lemma.Above estimate is proved using Gronwall’s lemma, Chebyshev’sinequality, and Garsia-Rodemick-Rumsey lemma ([Mo], Pitman
3
Books, 1984, Theorem 4.1, p. 150; [Mo], Pitman Books, 1984,Theorem 4.4, pp.152-154.)
(ii) By mean-square Lipschitz dependence ([Mo], Pitman Books,1984, Theorem 3.1, p. 41), the trajectory
[0, a]× C → L2(Ω, C) ⊂ L0(Ω, C)
(t, η) 7→ ηxt
is globally Lipschitz in η uniformly wrt t in compact sets, and iscontinuous in t for fixed η. Therefore it is jointly continuous in(t, η) as a map
[0, a]× C 3 (t, η) 7→ ηxt ∈ L0(Ω, C).
Then apply the Cohn-Hoffman-Jφrgensen theorem:If T, E are complete separable metric spaces, then each Borel map X : T →L0(Ω, E;F) admits a measurable version
T × Ω → E
to the trajectory field to get measurability in (t, η). (Take T =[0, a]× C, E = C ([Mo], Pitman Books, 1984, p. 16).)
(iii) Use the estimate
P(‖η1xt − η2xt‖Cα ≥ N
) ≤ C2k
N2k‖η1 − η2‖2k
C
for t ∈ [r, a], N > 0, ([Mo], 1984, Theorem 4.7, pp.158-162) toprove joint continuity of the trajectory
[r, a]× C → L0(Ω, Cα)
(t, η) 7→ ηxt
4
([Mo], Theorem 4.7, pp. 158-162) viewed as a process withvalues in the separable Banach space Cα. Again apply the Cohn-Hoffman-Jφrgensen theorem. ¤
As we have seen in Lecture I, the trajectory of a sfde possessesgood regularity properties in the mean-square. The following theoremshows good behavior in distribution.
Theorem III.2. ([Mo], Pitman Books, 1984)Suppose the coefficients H,G are globally Lipschitz in the second variable
uniformly with respect to the first. Let α ∈ (0, 1/2) and k be any integer such that
k > (1− 2α)−1. Then there are positive constants C3k , C4
k , C5k such that
dC(η1xt,η2xt) ≤ C3
k‖η1 − η2‖2k/(2k+1)C t ∈ [0, a]
dCα(η1xt,η2xt) ≤ C4
k‖η1 − η2‖2k/(2k+1)C t ∈ [r, a]
P(‖ηxt‖Cα ≥ N
)≤ C5
k(1 + ‖η‖2kC )
1N2k
, t ∈ [r, a], N > 0.
In particular the transition probabilities
[r, a]× C →Mp(C)
(t,η) 7→ p(0, η, t, ·)
take bounded sets into relatively weak* compact sets in the space Mp(C) of prob-
ability measures on C.
Proof of Theorem III.2.
Proofs of the estimates use Gronwall’s lemma, Chebyshev’s in-equality, and Garsia-Rodemick-Rumsey lemma ([Mo], 1984, Theorem4.1, p. 150; [Mo], 1984, Theorem 4.7, pp.159-162.) The weak* com-pactness assertion follows from the last estimate, Prohorov’s theorem
5
and the compactness of the embedding Cα → C ([Mo], 1984, Theorem4.6, pp. 156-158). ¤Erratic Behavior. The Noisy Loop Revisited
Definition.
A sfde is regular with respect to M2 if its trajectory random field(x(t), xt) : (x(0), x0) = (v, η) ∈ M2, t ≥ 0 admits a (Borel R+ ⊗ Borel M2 ⊗F , Borel M2)-measurable version X : R+×M2×Ω → M2 with a.a. samplefunctions continuous on R+ × M2. The sfde is said to be singularotherwise. Similarly for regularity with respect to C.
Consider the one-dimensional linear sdde with a positive delayr
dx(t) = σx(t− r)) dW (t), t > 0
(x(0), x0) = (v, η) ∈ M2 := R× L2([−r, 0],R),
(I)
driven by a Wiener process W .Theorem III.3 below implies that (I) is singular with respect to
M2 (and C).(See also [Mo], Stochastics, 1986).Consider the regularity of the more general one-dimen-sional
linear sfde:
dx(t) =∫ 0
−r
x(t + s)dν(s) dW (t), t > 0
(x(0), x0) ∈ M2 := R× L2([−r, 0],R)
(II ′)
where W is a Wiener process and ν is a fixed finite real-valued Borelmeasure on [−r, 0].Exercise:
(II′) is regular if ν has a C1 (or even L21) density with respect
to Lebesgue measure on [−r, 0]. ( Hint: Use integration by parts toeliminate the Ito integral!)
6
The following theorem gives conditions on the measure ν underwhich (II′) is singular.
Theorem III.3 ([M-S], II, 1996)Let r > 0, and suppose that there exists ε ∈ (0, r) such that supp ν ⊂
[−r,−ε]. Suppose 0 < t0 ≤ ε. For each k ≥ 1, set
νk :=√
t0
∣∣∣∣∫
[−r,0]
e2πiks/t0 dν(s)∣∣∣∣.
Assume that ∞∑
k=1
νkx1/ν2k = ∞ (1)
for all x ∈ (0, 1). Let Y : [0, ε] ×M2 × Ω → R be any Borel-measurable version of
the solution field x(t) : 0 ≤ t ≤ ε, (x(0), x0) = (v, η) ∈ M2 of (II′). Then for a.a.
ω ∈ Ω, the map Y (t0, ·, ω) : M2 → R is unbounded in every neighborhood of every
point in M2, and (hence) non-linear.
7
Corollary. ([Mo], Pitman Books, 1984 )Suppose r > 0, σ 6= 0 in (I). Then the trajectory ηxt : 0 ≤ t ≤ r, η ∈ C of
(I) has a measurable version X : R+ × C × Ω → C s.t. for every t ∈ (0, r]
P
(X(t, η1 + λη2, ·) = X(t, η1, ·)+λX(t, η2, ·)
for all λ ∈ R, η1, η2 ∈ C
)= 0.
But
P
(X(t, η1 + λη2, ·) = X(t, η1, ·) + λX(t, η2, ·)
)= 1.
for all λ ∈ R, η1, η2 ∈ C.
Remark.
(i) Condition (1) of the theorem is implied by
limk→∞
νk
√log k = ∞.
(ii) For the delay equation (I), ν = σδ−r, ε = r. In this case condition(1) is satisfied for every t0 ∈ (0, r].
(iii) Theorem III.3 also holds for state space C since every bound-edset in C is also bounded in L2([−r, 0],R).
8
Proof of Theorem III.3.
Joint work with V. Mizel.Main idea is to track the solution random field of (a complexified
version of) (II′) along the classical Fourier basis
ηk(s) = e2πiks/t0 , −r ≤ s ≤ 0, k ≥ 1 (2)
in L2([−r, 0],C). On this basis, the solution field gives an infinite familyof independent Gaussian random variables. This allows us to showthat no Borel measurable version of the solution field can be boundedwith positive probability on an arbitrarily small neighborhood of 0 inM2, and hence on any neighborhood of any point in M2 (cf. [Mo], Pit-man Books, 1984; [Mo], Stochastics, 1986). For simplicity of compu-tations, complexify the state space in (II′) by allowing (v, η) to belongto MC
2 := C× L2([−r, 0],C). Thus consider the sfde
dx(t) =∫
[−r,0]
x(t + s)dν(s) dW (t), t > 0,
(x(0), x0) = (v, η) ∈ MC2
(II ′ − C))
where x(t) ∈ C, t ≥ −r, and ν, W are real-valued.Use contradiction. Let Y : [0, ε] × M2 × Ω → R be any Borel-
measurable version of the solution field x(t) : 0 ≤ t ≤ ε, (x(0), x0) =(v, η) ∈ M2 of (II′). Suppose, if possible, that there exists a set Ω0 ∈ Fof positive P -measure, (v0, η0) ∈ M2 and a positive δ such that for allω ∈ Ω0, Y (t0, ·, ω) is bounded on the open ball B((v0, η0), δ) in M2 of center(v0, η0) and radius δ. Define the complexification Z(·, ω) : MC
2 → C ofY (t0, ·, ω) : M2 → R by
Z(ξ1 + iξ2, ω) := Y (t0, ξ1, ω) + i Y (t0, ξ2, ω), i =√−1,
for all ξ1, ξ2 ∈ M2, ω ∈ Ω. Let (v0, η0)C denote the complexification(v0, η0)C := (v0, η0) + i(v0, η0). Clearly Z(·, ω) is bounded on the complex
9
ball B((v0, η0)C , δ) in MC2 for all ω ∈ Ω0. Define the sequence of complex
random variables Zk∞k=1 by
Zk(ω) := Z((ηk(0), ηk), ω)− ηk(0), ω ∈ Ω, k ≥ 1.
ThenZk =
∫ t0
0
∫
[−r,−ε]
ηk(u + s) dν(s) dW (u), k ≥ 1.
By standard properties of the Ito integral, and Fubini’s theorem,
EZkZl =∫
[−r,−ε]
∫
[−r,−ε]
∫ t0
0
ηk(u + s)ηl(u + s′) du dν(s) dν(s′) = 0
for k 6= l, because ∫ t0
0
ηk(u + s)ηl(u + s′) du = 0
whenever k 6= l, for all s, s′ ∈ [−r, 0]. Furthermore∫ t0
0
ηk(u + s)ηk(u + s′) du = t0e2πik(s−s′)/t0
for all s, s′ ∈ [−r, 0]. Hence
E|Zk|2 =∫
[−r,−ε]
∫
[−r,−ε]
t0e2πik(s−s′)/t0 dν(s) dν(s′)
= t0
∣∣∣∣∫
[−r,0]
e2πiks/t0 dν(s)∣∣∣∣2
= ν2k .
Z(·, ω) : MC2 → C is bounded on B((v0, η0)C , δ) for all ω ∈ Ω0, and
‖(ηk(0), ηk)‖ =√
r + 1 for all k ≥ 1. By the linearity property
Z
((v0, η0)C+
δ
2√
r + 1(ηk(0), ηk), ·
)
= Z((v0, η0)C , ·) +δ
2√
r + 1Z((ηk(0), ηk), ·), k ≥ 1,
10
a.s., it follows thatP
(supk≥1
|Zk| < ∞)> 0. (3)
It is easy to check that ReZk, ImZk : k ≥ 1 are independentN (0, ν2
k/2)-distributed Gaussian random variables. Get a contradictionto (3):
For each integer N ≥ 1,
P
(supk≥1
|Zk| < N
)≤
∏
k≥1
P
(|ReZk| < N
)
=∏
k≥1
[1− 2√
2π
∫ ∞√
2Nνk
e−x2/2 dx
]
≤ exp− 2√
2π
∞∑
k=1
∫ ∞√
2Nνk
e−x2/2 dx
. (4)
There exists N0 > 1 (independent of k ≥ 1) such that∫ ∞√
2Nνk
e−x2/2 dx ≥ νk
2√
2Ne−N2
ν2k (5)
for all N ≥ N0 and all k ≥ 1.Combine (4) and (5) and use hypothesis (1) of the theorem to
getP
(supk≥1
|Zk| < N
)= 0
for all N ≥ N0. Hence
P
(supk≥1
|Zk| < ∞)
= 0.
This contradicts (3)(cf. Dudley [Du], JFA, 1967).Since Y (t0, ·, ω) is locally unbounded, it must be non-linear be-
cause of Douady’s Theorem:Every Borel measurable linear map between two Banach spaces is continuous.
(Schwartz [Sc], Radon Measures, Part II, 1973, pp. 155-160). ¤11
Note that the pathological phenomenon in Theorem III.3 is pe-culiar to the delay case r > 0 . The proof of the theorem suggests thatthis pathology is due to the Gaussian nature of the Wiener processW coupled with the infinite-dimensionality of the state space M2. Be-cause of this, one may expect similar difficulties in certain types oflinear spde’s driven by multi-dimensional white noise (Flandoli andSchaumloffel [F-S], Stochastics, 1990).Problem.
Classify all finite signed measures ν on [−r, 0] for which (II ′) isregular.
Note that (I) automatically satisfies the conditions of TheoremIII.3, and hence its trajectory field explodes on every small neighbor-hood of 0 ∈ M2. Because of the singular nature of (I), it is surprisingthat the maximal exponential growth rate of the trajectory of (I) isnegative for small σ and is bounded away from zero independently ofthe choice of the initial path in M2. This will be shown later in LectureV (Theorem V.1).
Regular Linear Systems. White Noise
SDE’s on Rd driven by m-dimensional Brownian motion W :=(W1, · · · , Wm), with smooth coefficients.
dx(t) = H(x(t− d1), · · · , x(t− dN ), x(t), xt)dt
+m∑
i=1
gix(t) dWi(t), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(V III)
(VIII) is defined on(Ω,F , (Ft)t≥0, P ) = canonical complete filtered Wiener space:
12
Ω := space of all continuous paths ω : R+ → Rm, ω(0) = 0, inEuclidean space Rm, with compact open topology;
F := completed Borel σ-field of Ω;Ft := completed sub-σ-field of F generated by the evaluations
ω → ω(u), 0 ≤ u ≤ t, t ≥ 0;P := Wiener measure on Ω;dWi(t) = Ito stochastic differentials, 1 ≤ i ≤ m.Several finite delays 0 < d1 < d2 < · · · < dN ≤ r in drift term; no
delays in diffusion coefficient.H : (Rd)N+1×L2([−r, 0],Rd) → Rd is a fixed continuous linear map;
gi, i = 1, 2, . . . , m, fixed (deterministic) d× d-matrices.Theorem III.4.([Mo], Stochastics, 1990])
(VIII) is regular with respect to the state space M2 = Rd × L2([−r, 0],Rd).There is a measurable version X : R+ × M2 × Ω → M2 of the trajectory field
(x(t), xt) : t ∈ R+, (x(0), x0) = (v, η) ∈ M2 with the following properties:
(i) For each (v, η) ∈ M2 and t ∈ R+, X(t, (v, η), ·) = (x(t), xt) a.s., is
Ft-measurable and belongs to L2(Ω,M2;P ).
(ii) There exists Ω0 ∈ F of full measure such that, for all ω ∈ Ω0, the
map X(·, ·, ω) : R+ ×M2 → M2 is continuous.
(iii) For each t ∈ R+ and every ω ∈ Ω0, the map X(t, ·, ω) : M2 → M2 is
continuous linear; for each ω ∈ Ω0, the map R+ 3 t 7→ X(t, ·, ω) ∈L(M2) is measurable and locally bounded in the uniform operator
norm on L(M2). The map [r,∞) 3 t 7→ X(t, ·, ω) ∈ L(M2) is contin-
uous for all ω ∈ Ω0.
(iv) For each t ≥ r and all ω ∈ Ω0, the map
X(t, ·, ω) : M2 → M2
is compact.
13
Proof uses variational technique to reduce the problem to thesolution of a random family of classical integral equations involvingno stochastic integrals.
Compactness of semi-flow for t ≥ r will be used later to definehyperbolicity for (VIII) and the associated exponential dichotomies(Lecture IV).
Regular Linear Systems. Semimartingale Noise
(Ω,F , (Ft)t≥0, P ) a complete filtered probability space satisfyingthe usual conditions.
Linear systems driven by semimartingale noise, and memorydriven by a measure-valued processν : R × Ω → M([−r, 0],Rd×d), where M([−r, 0],Rd×d) is the space of alld×d-matrix-valued Borel measures on [−r, 0] (or Rd×d-valued functionsof bounded variation on [−r, 0]). This space is given the σ-algebragenerated by all evaluations. The space Rd×d of all d × d-matrices isgiven the Euclidean norm ‖ · ‖.
dx(t) =∫
[−r,0]
ν(t)(ds)x(t + s)
dt + dN(t)∫ 0
−r
K(t)(s)x(t + s) ds
+ dL(t)x(t−), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(IX)
Hypotheses (R)
(i) The process ν : R×Ω →M([−r, 0],Rd×d) is measurable and (Ft)t≥0)-adapted. For each ω ∈ Ω and t ≥ 0 define the positive measureν(t, ω) on [−r,∞) by
ν(t, ω)(A) := |ν|(t, ω)(A− t) ∩ [−r, 0]
14
for all Borel subsets A of [−r,∞), where |ν| is the total variationmeasure of ν wrt the Euclidean norm on Rd×d. Therefore theequation
µ(ω)(·) :=∫ ∞
0
ν(t, ω)(·) dt
defines a positive measure on [−r,∞). For each ω ∈ Ω supposethat µ(ω) has a density wrt Lebesgue measure which is locallyessentially bounded.(Exercise: This condition is automatically satisfied if ν(t, ω) isindependent of (t, ω).)
(ii) K : R × Ω → L∞([−r, o],Rd×d) is measurable and (Ft)t≥0- adapted.Define the random field K(t, s, ω) byK(t, s, ω) := K(t, ω)(s − t) for t ≥ 0, −r ≤ s − t ≤ 0. Assume thatK(t, s, ω) is absolutely continuous in t for Lebesgue a.a. s and
all ω ∈ Ω. For every ω ∈ Ω, ∂K
∂t(t, s, ω) and K(t, s, ω) are locally
essentially bounded in (t, s). ∂K
∂t(t, s, ω) is jointly measurable.
(iii) L = M + V , M continuos local martingale, V B.V. process.Theorem III.5. ([M-S], I, AIHP, 1996)
Under hypotheses (R), equation (IX) is regular w.r.t. M2 with a measurable
flow X : R+ ×M2 × Ω → M2. This flow satisfies Theorem III.4.
Proof.
This is achieved via a construction in ([M-S], I, AIHP, 1996)which reduces (IX) to a random linear integral equation with no sto-chastic integrals ([M-S], AIHP, 1996, pp. 85-96). Do a complicatedpathwise analysis on the integral equation to establish existence andregularity properties of the semiflow. ¤
Regular Non-linear Systems
(a) SFDE’s with Ordinary Diffusion Coefficients15
In the sfde,
dx(t) = H(xt)dt +m∑
i=1
gi(x(t))dWi(t)
x0 = η ∈ C
(XV )
let H : C → Rd be globally Lipschitz and gi : Rd → Rd C2-bounded mapssatisfying the Frobenius condition (vanishing Lie brackets):
Dgi(v)gj(v) = Dgj(v)gi(v), 1 ≤ i, j ≤ m, v ∈ Rd;
and W := (W1,W2, · · · ,Wm) is m-dimensional Brownian motion. Notethat the diffusion coefficient in (XV) has no memory.
Theorem III.6 ([Mo], Pitman Books, 1984)Suppose the above conditions hold. Then the trajectory field ηxt : t ≥
0, η ∈ C of (XV) has a measurable version X : R+ × C × Ω → C satisfying the
following properties. For each α ∈ (0, 1/2), there is a set Ωα ⊂ Ω of full measure
such that for every ω ∈ Ωα
(i) X(·, ·, ω) : R+ × C → C is continuous;
(ii) X(·, ·, ω) : [r,∞)× C → Cα is continuous;
(iii) for each t ≥ r, X(t, ·, ω) : C → C is compact;
(iv) for each t ≥ r, X(t, ·, ω) : C → Cα is Lipschitz on every bounded set in
C, with a Lipschitz constant independent of t in compact sets. Hence each
map X(t, ·, ω) : C → C is compact: viz. takes bounded sets into relatively
compact sets.
16
Proof of Theorem III.6.
([Mo], Pitman Books, 1984, Theorem (2.1), Chapter (V), §2, p.121). This latter result is proved using a non-linear variational methodoriginally due to Sussman ([Su], Ann. Prob., 1978) and Doss ([Do],AIHP, 1977) in the non-delay case r = 0. Write g := (g1, g2, · · · , gm) :Rd → Rd×m. By the Frobenius condition, there is a C2 map F : Rm ×Rd → Rd such that F (t, ·) : t ∈ Rm is a group of C2 diffeomorphismsRd → Rd satisfying
D1F (t, x) = g(F (t, x)),
F (0, x) = x
for all t ∈ Rm, x ∈ Rd.Define
W 0(t) :=
W (t)−W (0), t ≥ 00 − r ≤ t < 0
and H : R+ × C × Ω → Rd, by
H(t, η, ·) :=D2F (W 0(t), η(0))−1H[F (W 0
t , η)]
− 12trace
(Dg[F (W 0(t), η(0))] g[F (W 0(t), η(0))]
)
where the expression under the “trace” is viewed as a bilinear formRm × Rm → Rd, and the trace has values in Rd. Then for each ω,H(t, η, ω) is jointly continuous, Lipschitz in η in bounded subsets of C
uniformly for t in compact sets, and satisfies a global linear growthcondition in η ([Mo], Pitman Books, 1984, pp. 114-126).
Therefore solve the fde
ηξ′t = H(t, ηξt, ·) t ≥ 0ηξ0 = η.
17
Define the semiflow
X(t, η, ω) = F (W 0
t (ω), ηxt(ω)).
Check that X satisfies all assertions of theorem ([Mo], 1984, pp.126-133). ¤
18
(b) SFDE’s with Smooth Memory
dx(t) = H(dt, x(t), xt) + G(dt, x(t), g(xt)), t > 0
(x(0), x0) = (v, eta) ∈ M2
(XV I)
Coefficients H and G in (XVI) are semimartingale-valued ran-dom fields on M2 = Rd × L2([−r, 0],Rd) andRd × Rm, respectively. The memory is driven by a functional g :L2([−r, 0],Rd) → Rm with the smoothness property that the processt 7→ g(xt) has absolutely continuous paths for each adapted process x.Under (technical) but general regularity and boundedness conditionson the characteristics of H and G, equation (XVI) is regular:
Theorem III.7 ([M-S], 1996)Let
∆ := (t0, t) ∈ R2 : t0 ≤ t.
Under suitable regularity conditions on H,G, g in (XVI), there exists a random field
X : ∆×M2 × Ω → M2 satisfying the following properties:
(i) For each (v, η) ∈ M2, (t0, t) ∈ ∆, X(t0, t, (v, η), ·) = (xt0,(v,η)(t), xt0,(v,η)t )
a.s., where xt0,(v,η) is the unique solution of (XVI) with xt0,(v,η)t0 =
(v, η).
(ii) For each (t0, t, ω) ∈ ∆× Ω, the map
X(t0, t, ·, ω) : M2 → M2
is C∞.
(iii) For each ω ∈ Ω and (t0, t) ∈ ∆ with t > t0 + r, the map
X(t0, t, ·, ω) : M2 → M2
carries bounded sets into relatively compact sets.
19
IV. ERGODIC THEORY OF REGULAR LINEAR SFDE’s
Geilo, Norway
Thursday, August 1, 1996
14:00-14:50
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
IV. ERGODIC THEORY OF LINEAR SFDE’s
1. Plan
Use state space M2. For regular linear sfde’s (VIII), (IX), con-sider the following themes:
I) Existence of a “perfect” cocycle on M2 that is a modification ofthe trajectory field (x(t), xt) ∈ M2.
II) Existence of almost sure Lyapunov exponents
limt→∞
1t
log ‖(x(t), xt)‖M2
The multiplicative ergodic theorem and hyperbolicity of the co-cycle.
III) The Stable Manifold Theorem, (viz. “random saddles”) for hy-perbolic systems.
2
2. Regular Linear Systems. White Noise
Linear sfde’s on Rd driven by m-dimensional Brownian motion
W := (W1, · · · ,Wm), with smooth coefficients.
dx(t) = H(x(t− d1), · · · , x(t− dN ), x(t), xt)dt
+m∑
i=1
gix(t) dWi(t), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(V III)
(VIII) is defined on
(Ω,F , (Ft)t∈R, P ) = canonical complete filtered Wiener space.
Ω := space of all continuous paths ω : R → Rm, ω(0) = 0, in
Euclidean space Rm, with compact open topology;
F := completed Borel σ-field of Ω;
Ft := completed sub-σ-field of F generated by the evaluations
ω → ω(u), u ≤ t, t ∈ R.
P := Wiener measure on Ω.
dWi(t) = Ito stochastic differentials.
Several finite delays 0 < d1 < d2 < · · · < dN ≤ r in drift term; no
delays in diffusion coefficient.
H : (Rd)N+1×L2([−r, 0],Rd) → Rd is a fixed continuous linear map,
gi, i = 1, 2, . . . , m, fixed (deterministic) d× d-matrices.
Recall regularity theorem:
Theorem III.4.([Mo], Stochastics, 1990])
3
(VIII) is regular with respect to the state space M2 = Rd × L2([−r, 0],Rd).
There is a measurable version X : R+ × M2 × Ω → M2 of the trajectory field
(x(t), xt) : t ∈ R+, (x(0), x0) = (v, η) ∈ M2 of (VIII) with the following proper-
ties:
(i) For each (v, η) ∈ M2 and t ∈ R+, X(t, (v, η), ·) = (x(t), xt) a.s., is
Ft-measurable and belongs to L2(Ω,M2;P ).
(ii) There exists Ω0 ∈ F of full measure such that, for all ω ∈ Ω0, the
map X(·, ·, ω) : R+ ×M2 → M2 is continuous.
(iii) For each t ∈ R+ and every ω ∈ Ω0, the map X(t, ·, ω) : M2 → M2 is
continuous linear; for each ω ∈ Ω0, the map R+ 3 t 7→ X(t, ·, ω) ∈L(M2) is measurable and locally bounded in the uniform operator
norm on L(M2). The map [r,∞) 3 t 7→ X(t, ·, ω) ∈ L(M2) is contin-
uous for all ω ∈ Ω0.
(iv) For each t ≥ r and all ω ∈ Ω0, the map
X(t, ·, ω) : M2 → M2
is compact.
Compactness of semi-flow for t ≥ r will be used below to define
hyperbolicity for (VIII) and the associated exponential dichotomies.
Lyapunov Exponents. Hyperbolicity
4
Version X of the flow constructed in Theorem III.4 is a multi-
plicative L(M2)-valued linear cocycle over the canonical Brownian shift
θ : R× Ω → Ω on Wiener space:
θ(t, ω)(u) := ω(t + u)− ω(t), u, t ∈ R, ω ∈ Ω.
Indeed we have
Theorem IV.1([M], 1990)
There is an F-measurable set Ω of full P -measure such that θ(t, ·)(Ω) ⊆ Ω
for all t ≥ 0 and
X(t2, ·, θ(t1, ω)) X(t1, ·, ω) = X(t1 + t2, ·, ω)
for all ω ∈ Ω and t1, t2 ≥ 0.
5
The Cocycle Property
6
Proof of Theorem IV.1. (Sketch)
For simplicity consider the case of a single delay d1; i.e. N = 1.
First step.
Approximate the Brownian motion W in (VIII) by smooth adapted
processes W k∞k=1:
W k(t) := k
∫ t
t−(1/k)
W (u) du− k
∫ 0
−(1/k)
W (u) du, t ≥ 0, k ≥ 1. (1)
Exercise: Check that each W k is a helix (i.e. has stationary incre-
ments):
W k(t1 + t2, ω)−W k(t1, ω) = W k(t2, θ(t1, ω)), t1, t2 ∈ R, ω ∈ Ω. (2)
Let Xk : R+ ×M2 ×Ω → M2 be the stochastic (semi)flow of the random
fde’s:
dxk(t) = H(xk(t− d1), xk(t), xkt )dt
+m∑
i=1
gix(t)(W ki )′(t) dt− 1
2
m∑
i=1
g2i xk(t) dt t > 0
(xk(0), xk0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(V III − k)
If X : R+ × M2 × Ω → M2 is the flow of (VIII) constructed in
Theorem III.4, then
limk→∞
sup0≤t≤T
‖Xk(t, ·, ω)−X(t, ·, ω)‖L(M2) = 0 (3)
for every 0 < T < ∞ and all ω in a Borel set Ω of full Wiener mea-
sure which is invariant under θ(t, ·) for all t ≥ 0 ([Mo], Stochastics,
7
1990). This convergence may be proved using the following stochastic
variational method:
Let φ : R+ × Ω → Rd×d be the d× d-matrix-valued solution of the
linear Ito sode (without delay):
dφ(t) =m∑
i=1
giφ(t) dWi(t) t > 0
φ(0, ω) = I ∈ Rd×d a.a. ω
(4)
Denote by φk : R+ × Ω → Rd×d, k ≥ 1, the d × d-matrix solution of the
random family of linear ode’s:
dφk(t) =m∑
i=1
giφk(t)(W k
i )′(t)− 12
m∑
i=1
g2i φk(t) dt t > 0
φk(0, ·) = I ∈ Rd×d.
(4′)
Let Ω be the sure event of all ω ∈ Ω such that
φ(t, ω) := limk→∞
φk(t, ω) (5)
exists uniformly for t in compact subsets of R+. Each φk is an Rd×d-
valued cocycle over θ, viz.
φk(t1 + t2, ω) = φk(t2, θ(t1, ω))φk(t1, ω) (6)
for all t1, t2 ∈ R+ and ω ∈ Ω. From the definition of Ω and passing to
the limit in (6) as k → ∞, conclude that φ(t, ω) : t > 0, ω ∈ Ω, is an
Rd×d-valued perfect cocycle over θ, viz.
(i) P (Ω) = 1;
8
(ii) θ(t, ·)(Ω) ⊆ Ω for all t ≥ 0;
(iii) φ(t1 + t2, ω) = φ(t2, θ(t1, ω))φ(t1, ω) for all
t1, t2 ∈ R+ and every ω ∈ Ω;
(iv) φ(·, ω) is continuous for every ω ∈ Ω.
Alternatively use the perfection theorem in ([M-S], AIHP, 1996,
Theorem 3.1, p. 79-82) for crude cocycles with values in a metrizable
second countable topological group. Observe that φ(t, ω) ∈ GL(Rd).
Define H : R+ ×Rd ×M2 × Ω → Rd by
H(t,v1, v, η, ω)
:= φ(t, ω)−1[H(φt(·, ω)(−d1, v1), φ(t, ω)(v), φt(·, ω) (idJ , η))] (7)
for ω ∈ Ω, t ≥ 0, v, v1 ∈ Rd, η ∈ L2([−r, 0],Rd), where
φt(·, ω)(s, v) =
φ(t + s, ω)(v) t + s ≥ 0
v −r ≤ t + s < 0
and
(idJ , η)(s) = (s, η(s)), s ∈ J.
Define Hk : R+ ×Rd ×M2 × Ω → Rd by a relation similar to (7) with φ
replaced by φk. Then the random fde’s
y′(t) = H(t, y(t− d1), y(t), yt, ω) t > 0
(y(0), y0) = (v, η) ∈ M2
(8)
9
yk′(t) = Hk(t, yk(t− d1), yk(t), ykt , ω) t > 0
(yk(0), yk0 ) = (v, η) ∈ M2
(9)
have unique non-explosive solutions
y, yk : [−r,∞)× Ω → Rd
([Mo], Stochastics, 1990, pp. 93-98). Ito’s formula implies that
X(t, v, η, ω) = (φ(t, ω)(y(t, ω)), φt(·, ω) (idJ , yt)) (10)
The chain rule gives a similar relation for Xk with φ replaced by φk
(Exercise; [Mo], Stochastics, 1990, pp. 96-97).
Get the convergence
limk→∞
|Hk(t, v1, v, η, ω)− H(t, v1, v, η, ω)| = 0 (11)
uniformly for (t, v1, v, η) in bounded sets of R+×Rd×M2. Use Gronwall’s
lemma and (11) to deduce (3).
Second step.
Fix ω ∈ Ω and use uniqueness of solutions to the approximat-
ing equation (VIII-k) and the helix property (2) of W k to obtain the
cocycle property for (Xk, θ):
Xk(t2, ·, θ(t1, ω)) Xk(t1, ·, ω) = Xk(t1 + t2, ·, ω)
for all ω ∈ Ω and t1, t2 ≥ 0, k ≥ 1.
10
Third step.
Pass to limit as k →∞ in the above identity and use the conver-
gence (3) in operator norm to get the perfect cocycle property for X.
¤
11
The a.s. Lyapunov exponents
limt→∞
1t
log ‖X(t, (v(ω), η(ω)), ω)‖M2 ,
(for a.a. ω ∈ Ω, (v, η) ∈ L2(Ω,M2)) of the system (VIII) are character-
ized by the following “spectral theorem”. Each θ(t, ·) is ergodic and
preserves Wiener measure P . The proof of Theorem IV.2 below uses
compactness of X(t, ·, ω) : M2 → M2, t ≥ r, together with an infinite-
dimensional version of Oseledec’s multiplicative ergodic theorem due
to Ruelle (1982).
Theorem IV.2. ([Mo], Stochastics, 1990)
Let X : R+ ×M2 × Ω → M2 be the flow of (VIII) given in Theorem III.4.
Then there exist
(a) an F-measurable set Ω∗ ⊆ Ω such that P (Ω∗) = 1 and
θ(t, ·)(Ω∗) ⊆ Ω∗ for all t ≥ 0,
(b) a fixed (non-random) sequence of real numbers λi∞i=1, and
(c) a random family Ei(ω) : i ≥ 1, ω ∈ Ω∗ of (closed) finite-codimensional
subspaces of M2, with the following properties:
(i) If the Lyapunov spectrum λi∞i=1 is infinite, then λi+1 < λi for
all i ≥ 1 and limi→∞
λi = −∞; otherwise there is a fixed (non-random)
integer N ≥ 1 such that λN = −∞ < λN−1 < · · · < λ2 < λ1;
(ii) each map ω 7→ Ei(ω), i ≥ 1, is F-measurable into the Grassman-
nian of M2;
12
(iii) Ei+1(ω) ⊂ Ei(ω) ⊂ · · · ⊂ E2(ω) ⊂ E1(ω) = M2, i ≥ 1, ω ∈ Ω∗;
(iv) for each i ≥ 1, codim Ei(ω) is fixed independently of ω ∈ Ω∗;
(v) for each ω ∈ Ω∗ and (v, η) ∈ Ei(ω)\Ei+1(ω),
limt→∞
1t
log ‖X(t, (v, η), ω)‖M2 = λi, i ≥ 1;
(vi) Top Exponent:
λ1 = limt→∞
1t
log ‖X(t, ·, ω)‖L(M2) for all ω ∈ Ω∗;
(vii) Invariance:
X(t, ·, ω)(Ei(ω)) ⊆ Ei(θ(t, ω))
for all ω ∈ Ω∗, t ≥ 0, i ≥ 1.
13
Spectral Theorem
14
Proof of Theorem IV.2 is based on Ruelle’s discrete version of
Oseledec’s multiplicative ergodic theorem in Hilbert space ([Ru], Ann.
of Math. 1982, Theorem (1.1), p. 248 and Corollary (2.2), p. 253):
Theorem IV.3 ([Ru], 1982)
Let (Ω,F , P ) be a probability space and τ : Ω → Ω a P -preserving trans-
formation. Assume that H is a separable Hilbert space and T : Ω → L(H) a
measurable map (w.r.t. the Borel field on the space of all bounded linear operators
L(H)). Suppose that T (ω) is compact for almost all ω ∈ Ω, and E log+ ‖T (·)‖ < ∞.
Define the family of linear operators Tn(ω) : ω ∈ Ω, n ≥ 1 by
Tn(ω) := T (τn−1(ω)) · · ·T (τ(ω)) T (ω)
for ω ∈ Ω, n ≥ 1.
Then there is a set Ω0 ∈ F of full P -measure such that
τ(Ω0) ⊆ Ω0, and for each ω ∈ Ω0, the limit
limn→∞
[Tn(ω)∗ Tn(ω)]1/(2n) := Λ(ω)
exists in the uniform operator norm and is a positive compact self-adjoint operator
on H. Furthermore each Λ(ω) has a discrete spectrum
eµ1(ω) > eµ2(ω) > eµ3(ω) > eµ4(ω) > · · ·
where the µi’s are distinct. If µi∞i=1 is infinite, then µi ↓ −∞; otherwise they
terminate at µN(ω) = −∞. If µi(ω) > −∞, then eµi(ω) has finite multiplicity
mi(ω) and finite-dimensional eigen-space Fi(ω), with mi(ω) := dimFi(ω). Define
E1(ω) := M2, Ei(ω) :=[⊕i−1
j=1Fj(ω)]⊥
, E∞(ω) := ker Λ(ω).
15
Then
E∞(ω) ⊂ · · · ⊂ Ei+1(ω) ⊂ Ei(ω) · · · ⊂ E2(ω) ⊂ E1(ω) = H
and
limn→∞
1n
log ‖Tn(ω)x‖H =
µi(ω), if x ∈ Ei(ω)\Ei+1(ω)−∞ if x ∈ ker Λ(ω).
Proof.
[Ru], Ann. of Math., 1982, pp. 248-254. ¤
The following “perfect” version of Kingman’s subadditive er-
godic theorem is also used to construct the shift invariant set Ω∗ ap-
pearing in Theorem IV.2 above.
16
Theorem IV.4([M, 1990])(“Perfect” Subadditive Ergodic Theorem)
Let f : R+ × Ω → R ∪ −∞ be a measurable process on the complete
probability space (Ω,F , P ) such that
(i) E sup0≤u≤1
f+(u, ·) < ∞, E sup0≤u≤1
f+(1− u, θ(u, ·)) < ∞;
(ii) f(t1 + t2, ω) ≤ f(t1, ω) + f(t2, θ(t1, ω)) for all t1, t2 ≥ 0 and every ω ∈ Ω.
Then there exist a setˆΩ ∈ F and a measurable f : Ω → R ∪ −∞ with the
properties:
(a) P ( ˆΩ) = 1, θ(t, ·)( ˆΩ) ⊆ ˆΩ for all t ≥ 0;
(b) f(ω) = f(θ(t, ω)) for all ω ∈ ˆΩ and all t ≥ 0;
(c) f+ ∈ L1(Ω,R;P );
(d) limt→∞
(1/t)f(t, ω) = f(ω) for every ω ∈ ˆΩ.
If θ is ergodic, then there exist f∗ ∈ R ∪ −∞ and ˜Ω ∈ F such that
(a)′ P ( ˜Ω) = 1, θ(t, ·)( ˜Ω) ⊆ ˜Ω, t ≥ 0;
(b)′ f(ω) = f∗ = limt→∞
(1/t)f(t, ω) for every ω ∈ ˜Ω.
Proof.
[Mo], Stochastics, 1990, Lemma 7, pp. 115–117. ¤
Proof of Theorem IV.2 is an application of Theorem IV.3. Re-
quires Theorem IV.4 and the following sequence of lemmas.
Lemma 1
17
For each integer k ≥ 1 and any 0 < a < ∞,
E sup0≤t≤a
‖φ(t, ω)−1‖2k < ∞;
E sup0≤t1,t2≤a
‖φ(t2, θ(t1, ·))‖2k < ∞.
Proof.
Follows by standard sode estimates, the cocycle property for φ
and Holder’s inequality. ([M], pp. 106-108). ¤
The next lemma is a crucial estimate needed to apply Ruelle-
Oseledec theorem (Theorem IV.3).
Lemma 2
E sup0≤t1,t2≤r
log+ ‖X(t2, ·, θ(t1, ·))‖L(M2) < ∞.
18
Proof.
If y(t, (v, η), ω) is the solution of the fde (8), then using Gronwall’s
inequality, taking E sup0≤t1,t2≤r
log+ sup‖(v,η)‖≤1
and applying Lemma 1, gives
E sup0≤t1,t2≤r
log+ sup‖(v,η)‖≤1
‖(y(t2, (v, η),θ(t1, ·)), yt2(·, (v, η), θ(t1, ·)))‖M2
< ∞.
Conclusion of lemma now follows by replacing ω′ with θ(t1, ω) in the
formula
X(t2, (v, η), ω′)
= (φ(t2, ω′)(y(t2, (v, η), ω′)), φt2(·, ω′) (idJ , yt2(·, (v, η), ω′))
and Lemma 1. ¤
The existence of the Lyapunov exponents is obtained by inter-
polating the discrete limit
1r
limk→∞
1k
log ‖X(kr, (v(ω), η(ω)), ω)‖M2 , (12)
a.a. ω ∈ Ω, (v, η) ∈ L2(Ω,M2), between delay periods of length r. This
requires the next two lemmas.
19
Lemma 3
Let h : Ω → R+ be F-measurable and suppose E sup0≤u≤r
h(θ(u, ·) is finite.
Then
Ω1 :=(
limt→∞
1th(θ(t, ·) = 0
)
is a sure event and θ(t, ·)(Ω1) ⊆ Ω1 for all t ≥ 0.
Proof.
Use interpolation between delay periods and the discrete ergodic
theorem applied to the L1 function
h := sup0≤u≤r
h(θ(u, ·).
([Mo], Stochastics, 1990, Lemma 5, pp. 111-113.) ¤
Lemma 4
Suppose there is a sure event Ω2 such that θ(t, ·)(Ω2) ⊆ Ω2 for all t ≥ 0, and
the limit (12) exists (or equal to −∞) for all ω ∈ Ω2 and all (v, η) ∈ M2. Then
there is a sure event Ω3 such that θ(t, ·)(Ω3) ⊆ Ω3 and
limt→∞
1t
log ‖X(t, (v, η), ω)‖M2 =1r
limk→∞
1k
log ‖X(kr, (v, η), ω)‖M2 , (13)
for all ω ∈ Ω3 and all (v, η) ∈ M2.
20
Proof:
Take Ω3 := Ω ∩ Ω1 ∩ Ω2. Use cocycle property for X, Lemma 2
and Lemma 3 to interpolate. ([Mo], Stochastics 1990, Lemma 6, pp.
113-114.) ¤
21
Proof of Theorem IV.2. (Sketch)
Apply Ruelle-Oseledec Theorem (Theorem IV.3) with
T (ω) := X(r, ω) ∈ L(M2), compact linear for ω ∈ Ω;
τ : Ω → Ω; τ := θ(r, ·).
Then cocycle property for X implies
X(kr, ω, ·) = T (τk−1(ω)) T (τk−2(ω)) · · · T (τ(ω)) T (ω)
:= T k(ω)
for all ω ∈ Ω.
Lemma 2 implies
E log+ ‖T (·)‖L(M2) < ∞.
Theorem IV.3 gives a random family of compact self-adjoint positive
linear operators Λ(ω) : ω ∈ Ω4 such that
limn→∞
[Tn(ω)∗ Tn(ω)]1/(2n) := Λ(ω)
exists in the uniform operator norm and is a positive compact operator
on M2 for ω ∈ Ω4, a (continuous) shift-invariant set of full measure.
Furthermore each Λ(ω) has a discrete spectrum
eµ1(ω) > eµ2(ω) > eµ3(ω) > eµ4(ω) > · · ·
where the µ′is are distinct, with no accumulation points except possibly
−∞. If µi∞i=1 is infinite, then µi ↓ −∞; otherwise they terminate at
22
µN(ω) = −∞. If µi(ω) > −∞, then eµi(ω) has finite multiplicity mi(ω) and
finite-dimensional eigen-space Fi(ω), with mi(ω) := dimFi(ω). Define
E1(ω) := M2, Ei(ω) :=[⊕i−1
j=1Fj(ω)]⊥
, E∞(ω) := ker Λ(ω).
Then
E∞(ω) ⊂ · · · ⊂ Ei+1(ω) ⊂ Ei(ω) · · · ⊂ E2(ω) ⊂ E1(ω) = M2.
Note that codimEi(ω) =∑i−1
j=1 mj(ω) < ∞. Also
limk→∞
1k
log ‖X(kr, (v, η), ω)‖M2 =
µi(ω), if (v, η) ∈ Ei(ω)\Ei+1(ω)−∞ if (v, η) ∈ ker Λ(ω).
The functions
ω 7→ µi(ω), ω 7→ mi(ω), ω 7→ N(ω)
are invariant under the ergodic shift θ(r, ·). Hence they take the fixed
values µi, mi, N almost surely, respectively.
Lemma 4 gives a continuous-shift-invariant sure event Ω∗ ⊆ Ω4
such that
limt→∞
1t
log ‖X(t, (v, η), ω)‖M2 =1r
limk→∞
1k
log ‖X(kr, (v, η), ω)‖M2
=µi
r=: λi,
for (v, η) ∈ Ei(ω)\Ei+1(ω), ω ∈ Ω∗, i ≥ 1.
λi :=µi
r: i ≥ 1 is the Lyapunov spectrum of (VIII).
23
Since Lyapunov spectrum is discrete with no finite accumulation
points, then λi : λi > λ is finite for all λ ∈ R.
To prove invariance of the Oseledec space Ei(ω) under the cocycle
(X, θ) use the random field
λ((v, η), ω) := limt→∞
1t
log ‖X(t, (v, η), ω)‖M2 (v, η) ∈ M2, ω ∈ Ω∗
and the relations
Ei(ω) := (v, η) ∈ M2 : λ((v, η), ω) ≤ λi,
λ(X(t, (v, η), ω), θ(t, ω)) = λ((v, η), ω), ω ∈ Ω∗, t ≥ 0
([Mo], Stochastics 1990, p. 122). ¤
24
The non-random nature of the Lyapunov exponents λi∞i=1 of
(VIII) is a consequence of the fact the θ is ergodic. (VIII) is said to
be hyperbolic if λi 6= 0 for all i ≥ 1. When (VIII) is hyperbolic the flow
satisfies a stochastic saddle-point property (or exponential dichotomy)
(cf. the deterministic case with E = C([−r, 0],Rd), gi ≡ 0, i = 1, . . . , m,
in Hale [H], Theorem 4.1, p. 181).
Theorem IV.5 (Random Saddles)([Mo], Stochastics, 1990)
Suppose the sfde (VIII) is hyperbolic. Then there exist
(a) a set Ω∗ ∈ F such that P (Ω∗) = 1, and θ(t, ·)(Ω∗) = Ω∗ for all t ∈ R,
and
(b) a measurable splitting
M2 = U(ω)⊕ S(ω), ω ∈ Ω∗,
with the following properties:
(i) U(ω), S(ω), ω ∈ Ω∗, are closed linear subspaces of M2, dim U(ω) is
finite and fixed independently of ω ∈ Ω∗.
(ii) The maps ω 7→ U(ω), ω 7→ S(ω) are F-measurable into the Grass-
mannian of M2.
(iii) For each ω ∈ Ω∗ and (v, η) ∈ U(ω) there exists τ1 = τ1(v, η, ω) > 0
and a positive δ1, independent of (v, η, ω) such that
‖X(t, (v, η), ω)‖M2 ≥ ‖(v, η)‖M2eδ1t, t ≥ τ1.
25
(iv) For each ω ∈ Ω∗ and (v, η) ∈ S(ω) there exists τ2 = τ2(v, η, ω) > 0
and a positive δ2, independent of (v, η, ω) such that
‖X(t, (v, η), ω)‖M2 ≤ ‖(v, η)‖M2e−δ2t, t ≥ τ2.
(v) For each t ≥ 0 and ω ∈ Ω∗,
X(t, ω, ·)(U(ω)) = U(θ(t, ω)),
X(t, ω, ·)(S(ω)) ⊆ S(θ(t, ω)).
In particular, the restriction
X(t, ω, ·) | U(ω) : U(ω) → U(θ(t, ω))
is a linear homeomorphism onto.
Proof.
[Mo], Stochastics, 1990, Corollary 2, pp. 127-130. ¤
26
The Stable Manifold Theorem
27
5. Regular Linear Systems. Helix Noise
dx(t) =∫
[−r,0]
ν(t)(ds)x(t + s)
dt + dN(t)∫ 0
−r
K(t)(s)x(t + s) ds
+ dL(t)x(t−), t > 0
(x(0), x0) = (v, η) ∈ M2 := Rd × L2([−r, 0],Rd)
(IX)
Linear systems driven by helix semimartingale noise, and memory
driven by a measure-valued process ν on a complete filtered probability
space (Ω,F , (Ft)t∈R, P ).
Hypotheses (C)
(i) The processes ν, K are stationary ergodic in the sense that
there is a measurable ergodic P -preserving flow θ : R×Ω →Ω such that for each t ∈ R, Ft = θ(t, ·)−1(F0) and
ν(t, ω) = ν(0, θ(t, ω)), t ∈ R, ω ∈ Ω
K(t, ω) = K(0, θ(t, ω)), t ∈ R, ω ∈ Ω.
(ii) L = M + V , M continuos local martingale, V B.V. process.
The processes N , L, M have jointly stationary ergodic in-
crements:
N(t + h, ω)−N(t, ω) = N(h, θ(t, ω)),
L(t + h, ω)− L(t, ω) = L(h, θ(t, ω)),
M(t + h, ω)−M(t, ω) = M(h, θ(t, ω)),
28
for t ∈ R, ω ∈ Ω.
Semimartingales satisfying Hypothesis (C)(ii) were studied by
de Sam Lazaro and P.A. Meyer ([S-M], 1971, 1976), Cinlar, Jacod,
Protter and Sharpe [CJPS], Protter [P], 1986.
Equation (IX) is regular w.r.t. M2 with a measurable flow X :
R+ ×M2 × Ω → M2. This flow satisfies Theorems III.4 and the cocycle
property. This is achieved via a construction in ([M-S], AIHP, 1996)
based on the following consequence of Hypothesis (C)(ii):
29
Theorem IV.6 ([Mo], Survey paper, 1992, [M-S], AIHP, 1996)
Suppose M satisfies Hypothesis (C)(ii). Then there is an (Ft)t≥0-adapted
version φ : R+ × Ω → Rd×d of the solution to the matrix equation
dφ(t) = dM(t)φ(t) t > 0
φ(0) = I ∈ Rd×d
(X)
and a set Ω1 ∈ F such that
(i) P (Ω1) = 1;
(ii) θ(t, ·)(Ω1) ⊆ Ω1 for all t ≥ 0;
(iii) φ(t1 + t2, ω) = φ(t2, θ(t1, ω))φ(t1, ω) for all
t1, t2 ∈ R+ and every ω ∈ Ω1;
(iv) φ(·, ω) is continuous for every ω ∈ Ω1.
A proof of Theorem IV.6 is given in ([Mo], Survey, 1992; [M-
S], AIHP, 1996): either by a double-approximation argument or via
perfection techniques.
The existence of a discrete non-random Lyapunov spectrum
λi∞i=1 for the sfde (IX) is proved via Ruelle-Oseledec multiplicative
ergodic theorem which requires the integrability property (Lemma 2):
E sup0≤t1,t2≤r
log+ ‖X(t1, θ(t2, ·), ·)‖L(M2) < ∞.
The above integrability property is established under the following set
of hypotheses on ν, K, N , L:
30
Hypotheses (I)
(i)
sup−r≤s≤2r
∣∣∣∣d¯ν(·)(s)
ds
∣∣∣∣2
, sup0≤t≤2r,−r≤s≤0
‖K(t, ·)(s)‖3,
sup0≤t≤2r,−r≤s≤0
‖ ∂
∂tK(t, ·)(s)‖3, sup
0≤t≤2r,−r≤s≤0‖ ∂
∂sK(t, ·)(s)‖3,
|V |(2r, ·)4,
are all integrable, where
¯ν(ω)(A) :=∫ ∞
0
|ν(t, ω)|(A− t) ∩ [−r, 0] dt, A ∈ Borel[−r,∞)
has a locally (essentially) bounded density d¯ν(·)(s)ds
; and |V | =
total variation of V w.r.t. the Euclidean norm ‖ · ‖ on Rd×d.
(ii) Let N = N0 +V 0 where the local (Ft)t≥0-martingale N0 = (N0ij)
di,j=1
and the bounded variation process
V 0 = (V 0ij)
di,j=1 are such that
[N0ij ](2r, ·)2, |V 0
ij |(2r, ·)4, i, j = 1, 2, . . . , d
are integrable.
|V 0ij |(2r, ·) = total variation of V 0
ij over [0, 2r].
(iii) [Mij ](1) ∈ L∞(Ω,R), i, j = 1, 2, . . . , d.
The integrability property of the cocycle (X, θ) is a consequence
of
E log+ sup0≤t1,t2≤r, ‖(v,η)‖≤1
|x(t1, (v, η), θ(t2, ·))| < ∞.
31
Proof of latter property uses lengthy argument based on establishing
the existence of suitable higher order moments for the coefficients of
an associated random integral equation. (See Lemmas (5.1)-(5.5) in
[M-S],I, AIHP, 1996.)
Since θ is ergodic, the multiplicative ergodic theorem ( Theorem
IV.3, Ruelle) now gives a fixed discrete set of Lyapunov exponents
Theorem IV.7 ([Mo], Survey, 1992; [M-S], AIHP, 1996)
Under Hypotheses (C) & (I), the statements of Theorems IV.2 and IV.5 hold
true for the linear sfde (IX).
Note that the Lyapunov spectrum of (IX) does not change if
one uses the state space D([−r, 0],Rd) with the supremum norm ‖ · ‖∞([M-S], AIHP 1996).
32
V. STABILITY
EXAMPLES AND CASE STUDIES
Geilo, Norway
Friday, August 2, 1996
14:00-14:50
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
V. STABILITY. EXAMPLES AND CASE STUDIES
1. Plan.
I) Estimates on the “maximal exponential growth rate” for thesingular noisy feedback loop. Use of Lyapunov functionals.
II) Examples and case studies of linear sfde’s: Existence of thestochastic semiflow and its Lyapunov spectrum.
III) Study almost sure asymptotic stability via upper bounds on thetop Lyapunov exponent λ1.
IV) Lyapunov spectrum for sdde’s with Poisson noise.
2
Lyapunov exponents for linear sode’s (without memory): stud-ied by many authors: e.g. Arnold, Kliemann and Oeljeklaus, 1989,Arnold, Oeljeklaus and Pardoux, 1986, Baxendale, 1985, Pardoux andWihstutz[ PW1], 1988, Pinsky and Wihstutz [PW2], 1988, and thereferences therein.
Asymptotic stability of sfde’s: treated in Kushner [K], JDE,1968, Mizel and Trutzer [MT],1984, Mohammed [M1]-[M4], 1984,1986, 1990, 1992, Mohammed and Scheutzow [MS], 1996, Scheut-zow [S], 1988, Kolmanovskii and Nosov [KN], 1986. Mao ([Ma], 1994,Chapter 5) gives several results concerning top exponential growthrate for sdde’s driven by C-valued semimartingales. Assumes thatsecond-order characteristics of the driving semimartingales are time-dependent and decay to zero exponentially fast in time, uniformly inthe space variable.
3
2. Noisy Feedback Loop Revisited Once More!
Noisy feedback loop is modelled by the one-dimensional linearsdde
dx(t) = σx(t− r)) dW (t), t > 0
(x(0), x0) = (v, η) ∈ M2 := R× L2([−r, 0],R),
(I)
driven by a Wiener process W with a positive delay r.(I) is singular with respect to M2 (Theorem III.3).Consider the more general one-dimensional linear sfde:
dx(t) =∫ 0
−r
x(t + s)dν(s) dW (t), t > 0
(x(0), x0) ∈ M2 := R× L2([−r, 0],R)
(II ′)
where W is a Wiener process and ν is a fixed finite real-valued Borelmeasure on [−r, 0].
(II ′) is regular if ν has a C1 (or even L21) density with respect
to Lebesgue measure on [−r, 0] ([M-S], I, 1996). If ν satisfies TheoremIII.3, then (II ′) is singular.
In the singular case, there is no stochastic flow (Theorem III.3)and we do not know whether a (discrete) set of Lyapunov exponents
λ((v, η), ·) := limt→∞
1t
log ‖(x(t, (v, η)), xt(·, (v, η)))‖M2 , (v, η) ∈ M2
exists. Existence of Lyapunov exponents for singular equations ishard. But can still define the maximal exponential growth rate
λ1 := sup(v,η)∈M2
lim supt→∞
1t
log ‖(x(t, (v, η)), xt(·, (v, η)))‖M2
for the trajectory random field (x(t, (v, η)), xt(·, (v, η))) : t ≥ 0, (v, η) ∈ M2.λ1 may depend on ω ∈ Ω. But λ1 = λ1 in the regular case.
4
Inspite of the extremely erratic dependence on the initial pathsof solutions of (I), it is shown in Theorem V.1 that for small noisevariance, uniform almost sure global asymptotic stability still persists.For small σ, λ1 ≤ −σ2/2 + o(σ2) uniformly in the initial path (TheoremV.1, and Remark (iii)). For large |σ| and ν = δ−r,
12r
log |σ|+ o(log |σ|) ≤ λ1 ≤ 1r
log |σ|
([M-S], II, 1996, Remark (ii) after proof ofTheorem 2.3 ). This resultis in sharp contrast with the non-delay case (r = 0), where λ1 = −σ2/2for all values of σ. Proofs of Theorems V.1, V.2 involve very delicateconstructions of new types of Lyapunov functionals on the underlyingstate space.
Theorem V.1.([M-S], II, 1996).Let ν be a probability measure on [−r, 0], r > 0, and consider the sfde
dx(t) = σ
(∫
[−r,0]
x(t + s) dν(s))
dW (t), t ≥ 0
(x(0), x0) = (v, η) ∈ M2
(II ′)
with σ ∈ R, (v, η) ∈ M2, W standard Brownian motion, and x(·, (v, η)) the solution
of (II ′) through (v, η) ∈ M2. Then there exists σ0 > 0 and a continuous strictly
negative nonrandom function φ : (−σ0, σ0) → R− (independent of (v, η) ∈ M2 and
ν) such that
P
(lim sup
t→∞1t
log ‖(x(t, (v, η)), xt(·, (v, η)))‖M2 ≤ φ(σ))
= 1.
for all (v, η) ∈ M2 and all −σ0 < σ < σ0.
Remark:
Theorem also holds for state space C with ‖ · ‖∞.
Proof of Theorem V.1. (Sketch)
5
Sufficient to consider (II ′) on C ≡ C([−r, 0],R), because C is con-tinuously embedded in M2. W.l.o.g., assume that σ > 0.
• Use Lyapunov functional V : C → R+
V (η) := (R(η) ∨ |η(0)|)α + βR(η)α, η ∈ C.
where R(η) := η− η, the diameter of the range of η, η := sup−r≤s≤0
η(s)
and η := inf−r≤s≤0
η(s).
• Fix 0 < α < 1 and arrange for β = β(σ) for sufficiently small σ
such thatE(V (ηxr)) ≤ δ(σ)V (η), η ∈ C, (1)
and δ(σ) ∈ (0, 1) is a continuous function of σ defined near 0.There is a positive K = K(α) (independent of η, ν) such thatδ(σ) ∼ (1−Kσ2). Set
φ(σ) :=1α
log δ(σ).
Estimate (1) is hard ([M-S], II, 1996, pp. 12-18).• ηxnr∞n=1 is a Markov process in C. So (1) implies that δ(σ)−nV (ηxnr), n ≥
1, is a non-negative (Fnr) supermartingale.• There exists Z : Ω → [0,∞) such that
limn→∞
V (ηxnr)δ(σ)n
= Z a.s. (2)
• Form of V and (2) imply
limt→∞
1t
log |x(t)| ≤ limn→∞
1nr
log[|x(nr)|+ R(xnr)]
=1α
limn→∞
1nr
log V (xnr) ≤ 1α
log δ(σ) = φ(σ) < 0.
6
• δ(σ), φ(σ) independent of η, ν. “Domain” of φ also independentof η, ν. ¤
Remarks.
(i) Choice of σ0 in Theorem V.1 depends on r. In (I) the scalingt 7→ t/r has the effect of replacing r by 1 and σ by σ
√r. If λ1(r, σ)
is the maximal exponential growth rate of (I), then λ1(r, σ) =1rλ1(1, σ
√r) (Exercise). Hence σ0 decreases (like 1√
r) as r increases
. Thus (for a fixed σ), a small delay r tends to stabilize equation(I). A large delay in (I) has a destabilizing effect (Theorem V.2below).
(ii) Using a Lyapunov function(al) argument, Theorem V.2 belowshows that for sufficiently large σ, the singular delay equation(I) is unstable. Result is in sharp contrast with the non-delaycase r = 0, where
limt→∞
1t
log |x(t)| = −σ2/2 < 0
for all σ ∈ R (even when σ is large).(iii) The growth rate function φ in Theorem V.1 satisfies
φ(σ) = −σ2/2 + o(σ2)
as σ → 0+. Agrees with non-delay case r = 0. Above relationfollows by modifying proof of Theorem V.1.
7
Theorem V.2.
Consider the equation
dx(t) = σx(t− r) dW (t), t > 0
(x(0), x0) = (v, η) ∈ M2 := R× L2([−r, 0],R),
(I)
driven by a standard Wiener process W with a positive delay r and σ ∈ R . Then
there exists a continuous function ψ : (0,∞) → R which is increasing to infinity
such that
P
(lim inft→∞
1t
log ‖(x(t, (v, η)), xt(·, (v, η))‖M2 ≥ ψ(|σ|))
= 1,
for all (v, η) ∈ M2\0 and all σ 6= 0. The function ψ is independent of the choice
of (v, η) ∈ M2\0.Remarks.
(i) ‖ · ‖M2 can be replaced by the sup-norm on C.(ii) Proof shows ψ(σ) ∼ 1
2 log σ for large σ.
8
Proof of Theorem V.2.
Use the continuous Lyapunov functional
V : M2\0 → [0,∞)
V ((v, η)) :=(
v2 + |σ|∫ 0
−1
η2(s) ds
)−1/4
[M-S], Part II, 1996, pp. 20-24. ¤3. Regular one-dimensional linear sfde’s
To outline a general scheme for obtaining estimates on the topLyapunov exponent for a class of one-dimensional regular linear sfde’s.Then apply scheme to specific examples within the above class.
Scheme applies to multidimensional linear equations with mul-tiple delays.
Note: Approach in ([Ku], JDE, 1968) uses Lyapunov functionalsand yields strictly weaker estimates in all cases.
Consider the class of one-dimensional linear sfde’s
dx(t) =
ν1x(t) + µ1x(t− r) +∫ 0
−r
x(t + s)σ1(s) ds
dt
+
ν2x(t) +∫ 0
−r
x(t + s)σ2(s) ds
dM(t),
(XV II)
where r > 0, σ1, σ2 ∈ C1([−r, 0],R), and M is a continuous helix localmartingale on (Ω,F , (Ft)t≥0, P ) with (stationary) ergodic increments.Ergodic theorem gives the a.s. deterministic limit β := lim
t→∞〈M〉(t)
t.
Assume that β < ∞ and 〈M〉(1) ∈ L∞(Ω,R) .Hence (XVII) is regular with respect to M2 and has a sample-
continuous stochastic semiflow X : R+ ×M2 ×Ω → M2 (Theorem III.5).The stochastic semiflow X has a fixed (non-random) Lyapunov spec-trum (Theorem IV.7). Let λ1 be its top exponent. We wish to develop
9
an upper bound for λ1. By the spectral theorem (Theorem IV.7, cf.Theorem IV.2), there is a shift-invariant set Ω∗ ∈ F of full P -measureand a measurable random field λ : M2 × Ω → R ∪ −∞,
λ((v, η), ω) := limt→∞
1t
log ‖X(t, (v, η), ω)‖M2 , (v, η) ∈ M2, ω ∈ Ω∗, (1)
giving the Lyapunov spectrum of (XVII).Introduce family of equivalent norms
‖(v, η)‖α :=
αv2 +∫ 0
−r
η(s)2 ds
1/2
, (v, η) ∈ M2, α > 0, (2)
on M2. Then
λ((v, η), ω) = limt→∞
1t
log ‖X(t, (v, η), ω)‖α, (v, η) ∈ M2, ω ∈ Ω∗ (3)
for all α > 0; i.e. the Lyapunov spectrum of (XVII) with respect to‖ · ‖α is independent of α > 0.
Let x be the solution of (XVII) starting at (v, η) ∈ M2. Define
ρα(t)2 := ‖X(t)‖2α = αx(t)2 +∫ t
t−r
x(u)2 du, t > 0, α > 0. (4)
For each fixed (v, η) ∈ M2, define the set Ω0 ∈ F by Ω0 := ω ∈ Ω :ρα(t, ω) 6= 0 for all t > 0. If P (Ω0) = 0, then by uniqueness there is arandom time τ0 such that a.s. X(t, (v, η), ·) = 0 for all t ≥ τ0. Henceλ1 = −∞. So suppose that P (Ω0) > 0. Ito’s formula implies
log ρα(t) = log ρα(0) +∫ t
0
Qα(a(u), b(u), I1(u)) du
+∫ t
0
Qα(a(u), I2(u)) d〈M〉(u) +∫ t
0
Rα(a(u), I2(u)) dM(u), (5)
10
for t > 0, a.s. on Ω0, where
Qα(z1, z2, z3) := ν1z21 +
√α µ1z1z2 +
√α z1z3 + 1
2
z21
α− 1
2z22
Qα(z1, z′3) := α( 1
2 − z21)
(ν2√α
z1 + z′3
)2
Rα(z1, z′3) := ν2z
21 +
√αz1z
′3, ‖σi‖2 :=
∫ 0
−r
σi(s)2ds
1/2
,
(6)
i = 1, 2, and
a(t) :=√
αx(t)ρα(t)
, b(t) :=x(t− r)ρα(t)
, Ii(t) :=
∫ 0
−rx(t + s)σi(s) ds
ρα(t)(7)
for i = 1, 2, t > 0, a.s. on Ω0.Since
|Ii(t)| ≤ 1ρα(t)
(∫ 0
−r
x(t + s)2 ds
)1/2
‖σi‖2 =√
1− a2(t) ‖σi‖2,
i = 1, 2, a.s. on Ω0 the variables z1, z2, z3, z′3 in (6) must satisfy
|z1| ≤ 1, z2 ∈ R, |z3|2 ≤ (1− z21)‖σ1‖22, |z′3|2 ≤ (1− z2
1)‖σ2‖22.
Let τ1 := inft > 0 : ρα(t) = 0. Then the local martingale∫ t∧τ1
0
Rα(a(u), I2(u)) dM(u), t > 0,
is a time-changed (possibly stopped) Brownian motion. Since |Rα(a(u), I2(u))| ≤|ν2|+
√α‖σ2‖2 for all u ∈ [0, τ1), a.s., then
limt→∞
1t
∫ t∧τ1
0
Rα(a(u), I2(u)) dM(u) = 0 a.s. (8)
11
Divide (5) by t, let t →∞, to get
λ((v, η), ω) ≤ lim supt→∞
1t
∫ t
0
Qα(a(u), b(u), I1(u)) du
+ lim supt→∞
1t
∫ t
0
Qα(a(u), I2(u)) d〈M〉(u). (9)
a.s. on Ω0, for all α > 0.Wish to develop upper bounds on λ1 in the following cases.One-dimensional linear sfde (smooth memory in white-noise term):
dx(t) = ν1x(t) + µ1x(t− r) dt +∫ 0
−r
x(t + s)σ2(s) ds
dW (t), t > 0 (V II)
with real constants ν1, µ1 and σ2 ∈ C1([−r, 0],R). It is a special caseof (XVII). Hence (VII) is regular with respect to M2. The process∫ 0
−rx(t + s)σ2(s) ds has C1 paths in t. Hence the stochastic differential
dW in (VII) may be interpreted in the Ito or Stratonovich sense withoutchanging the solution x.
Theorem V.3.
Suppose λ1 is the top a.s. Lyapunov exponent of (VII). Define the function
θ(δ, α) := −δ +(
ν1 + δ +12αµ2
1e2δr +
12α
)∨
(α
2‖σ2‖22e2δ+r
)
for all α ∈ R+, δ ∈ R, where δ+ := maxδ, 0.Then
λ1 ≤ infθ(δ, α) : δ ∈ R, α ∈ R+. (10)
Proof.
Maximize the integrand on the right-hand-side of (9) (with M =W); then use exponential shift by δ to refine the resulting estimate.Then minimize over α, δ ([M-S], II, 1996, pp. 34-35). ¤
12
Corollary below shows that the estimate in Theorem V.3 reducesto well-known estimate in deterministic case σ2 ≡ 0 (Hale [Ha], pp.17-18).Corollary V.3.1.
In (VII), suppose µ1 6= 0 and let δ0 be the unique real solution of the tran-
scendental equation
ν1 + δ + |µ1|eδr = 0. (11)
Then
λ1 ≤ −δ0 +12‖σ2‖22|µ1| e|δ0|r. (12)
If µ1 = 0 and ν1 ≥ 0, then λ1 ≤ 12
(ν1 +
√ν21 + ‖σ2‖22
). If µ1 = 0 and ν1 < 0, then
λ1 ≤ ν1 + 12‖σ2‖2e−ν1r.
Proof.
Suppose µ1 6= 0. Denote by f(δ), δ ∈ R, the left-hand-side of (11).Then f(δ) is an increasing function of δ. f has a unique real zero δ0.Using (10), we may put δ = δ0 and α = |µ1|−1e−δ0r in the expression forθ(δ, α). This gives (12).
Suppose µ1 = 0. Put δ = (−ν1)+ in θ(δ, α) and minimize the re-sulting expression over all α > 0. This proves the last two assertionsof the corollary ([M-S], II, 1996, pp. 35-36). ¤Remarks.
(i) Upper bounds for λ1 in Theorem (V.3) and Corollary V.3.1 agreewith corresponding bounds in the deterministic case (for µ1 ≥ 0),but are not optimal when µ1 = 0 and σ2 is strictly positive andsufficiently small; cf. Theorem V.1 for small ‖σ2‖2.
(ii) Problem: What are the asymptotics of λ1 for small delays r ↓ 0?
Our second example is the stochastic delay equation
dx(t) = ν1x(t) + µ1x(t− r) dt + x(t)dM(t), t > 0, (XV III)
13
where M is the helix local martingale appearing in (XVII) and satis-fying the conditions therein. Hence (XVIII) is regular with respect toM2. Theorem below gives estimate on its top exponent.
Theorem V.4.
In (XVIII) define δ0 as in Corollary V.3.1. Then the top a.s. Lyapunov
exponent λ1 of (XVIII) satisfies
λ1 ≤ −δ0 +β
16. (13)
Proof.
Maximize the following functions separately over their appro-priate ranges:
Qα(z1, z2) := ν1z21 +
√α µ1z1z2 + 1
2
z21
α− 1
2z22 ,
Qα(z1) := ( 12 − z2
1)z21 , |z1| ≤ 1, z2 ∈ R.
Then use an exponential shift of the Lyapunov spectrum by an amountδ. Minimize the resulting bound over all α (for fixed δ) and then overall δ ∈ R. This minimum is attained if δ solves the transcendentalequation (11). Hence the conclusion of the theorem ([M-S], II, 1996,pp. 36-37). ¤Remark.
The above estimate for λ1 is sharp in the deterministic caseβ = 0 and µ1 ≥ 0, but is not sharp when β 6= 0; e.g. M = W ,one-dimensional standard Brownian motion in the non-delaycase (µ1 = 0). When M = ν2W for a fixed real ν2, the abovebound may be considerably sharpened as in Theorem V.5 be-low. The sdde in this theorem is a model of dye circulation in
14
the blood stream (cf. Bailey and Williams [B-W], 1996; Lenhartand Travis, 1986).
Theorem V.5.([M-S], II, 1996).
For the equation
dx(t) = ν1x(t) + µ1x(t− r)dt + ν2x(t) dW (t) (V I)
set
φ(δ) := −δ +1
4ν22
[(|µ1|eδr + ν1 + δ +
12ν22
)+]2
, (14)
for ν2 6= 0. Then
λ1 ≤ infδ∈R
φ(δ). (15)
In particular, if δ0 is the unique solution of the equation
ν1 + δ + |µ1|eδr +12ν22 = 0, (16)
then λ1 ≤ −δ0.
Proof.
Maximize
Qα(z1, z2, 0) + Qα(z1, 0) =(
ν1 +12α
+ν22
2
)z21 +
√α µ1z1z2 − 1
2z22 − ν2
2z41 (17)
for |z1| ≤ 1, z2 ∈ R and then minimize the resulting bound for λ1 overα > 0. Get
λ1 ≤ 116ν2
2
[(2ν1 + 2|µ1|+ ν2
2)+]2
.
The first assertion of the theorem follows from above estimate byapplying an exponential shift to (VI). Last assertion of the theoremis obvious ([M-S], II, 1996, pp. 38-39.) ¤Problem: Is λ1 = infδ∈R φ(δ) ?
15
Remark.
Estimate in Theorem V.5 agrees with the non-delay case µ1 = 0whereby λ1 = ν1− 1
2ν22 = inf
δ∈Rφ(δ). Cf. also [AOP], 1986, [B], 1985,
and [AKO], 1989.4. SDDE with Poisson Noise.
Consider the one-dimensional linear delay equation
dx(t) = x((t− 1)−) dN(t) t > 0
x0 = η ∈ D := D([−1, 0],R).
(V )
The process N(t) ∈ R is a Poisson process with i.i.d. inter-arrival timesTi∞i=1 which are exponentially distributed with the same parameterµ. The jumps Yi∞i=1 of N are i.i.d. and independent of all the Ti’s.Let
j(t) := sup
j ≥ 0 :j∑
i=1
Ti ≤ t
.
Then
N(t) =j(t)∑
i=1
Yi.
Equation (V) can be solved a.s. in forward steps of lengths 1, usingthe relation
xη(t) = η(0) +j(t)∑
i=1
Yix
(( i∑
j=1
Tj − 1)−
)a.s.
Trajectory xt : t ≥ 0 is a Markov process in the state space D
(with the supremum norm ‖ · ‖∞). Furthermore, the above relationimplies that (V) is regular in D; i.e., it admits a measurable flowX : R+ ×D × Ω → D with X(t, ·, ω) = ηxt(·, ω), continuous linear in η forall t ≥ 0 and a.a. ω ∈ Ω (cf. the singular equation (I) ).
16
The a.s. Lyapunov spectrum of (V) may be characterized di-rectly (without appealing to the Oseledec Theorem) by interpolatingbetween the sequence of random times:
τ0(ω) := 0,
τ1(ω) := inf
n ≥ 1 :k∑
j=1
Tj /∈ [n− 1, n] for all k ≥ 1
,
τi+1(ω) := inf
n > τi(ω) :k∑
j=1
Tj /∈ [n− 1, n] for all k ≥ 1
, i ≥ 1.
It is easy to see that τ1, τ2 − τ1, τ3 − τ2, · · · are i.i.d. and Eτ1 = eµ.
Theorem V.6.([M-S], II, 1996)
Let ξ ∈ D be the constant path ξ(s) = 1 for all s ∈ [−1, 0]. Suppose
E log ‖X(τ1(·), ξ, ·)‖∞ exists (possibly = +∞ or −∞). Then the a.s. Lyapunov
spectrum
λ(η) := limt→∞
1t
log ‖X(t, η, ω)‖∞, η ∈ D, ω ∈ Ω
of (V) is −∞, λ1 where
λ1 = e−µ E log ‖X(τ1(·), ξ, ·)‖∞.
In fact,
limt→∞
1t
log ‖X(t, η, ω)‖∞ =
λ1 η /∈ Ker X(τ1(ω), ·, ω)−∞ η ∈ Ker X(τ1(ω), ·, ω).
Proof.
The i.i.d. sequence
Si :=‖(X(τi, ξ, ·))‖‖(X(τi−1, ξ, ·))‖ i = 1, 2, . . .
17
and the LLN give
limn→∞
1τn
log ‖(X(τn, ξ, ω))‖ = e−µ(E log S1)
for a.a. ω ∈ Ω.Interpolate between the times τ1, τ2, τ3, · · · to get the continuos
limit ([M-S], II, 1996, pp. 27-28). ¤
18
VI. MISCELLANY
Geilo, Norway
Saturday, August 3, 1996
14:00-14:50
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
VI. MISCELLANY
1. Malliavin Calculus of SFDE’s
Objectives.
I) To establish the existence of smooth densities for solutions ofRd-valued sfde’s of the form
dx(t) = H(t, xt) dt + g(t, x(t− r)) dW (t). (XIX)
In the above equation, W is an m-dimensional Wiener process, r
is a positive time delay, H is a functional C([−r, 0],Rd) → Rd andg : [0,∞)×Rd → Rd×m is a d×m-matrix-valued function, such thatfor fixed t thed×d-matrix g(t, x)g(t, x)∗ has degeneracies of polynomial order asx runs on a hypersurface in Rd.
II) Method of proof gives a very general criterion for the hypoel-lipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent princi-pal part.
III) More generally the analysis works when H is replaced by a non-anticipating functional which may depend on the whole historyof the path ([B-M], Ann. Prob., 1995).
Case H ≡ 0 studied by (Bell and Mohammed [B-M], (J.F.A.,1991). Solution x(t) has smooth density wrt Lebesgue measure on Rd
if g(t, x)g(t, x)∗ degenerates like |x|2 near 0 (e.g. g(t, ·) linear.) Proof usesMalliavin calculus.
Difficulties.
(i) The infinitesimal generator of the trajectory Feller process xt :t ≥ 0 is a highly degenerate second-order differential operator
2
on the state space: its principal part degenerates on a surfaceof finite codimension (Lecture II, Theorem II.3). Hence cannotuse existing techniques from pde’s.
(ii) Analysis by Malliavin calculus requires derivation of probabilis-tic lower bounds on the Malliavin covariance matrix of the solu-tion x. These bounds are difficult because there is no stochasticflow in the singular case (Lecture III, Theorem III.3). cf. sodecase, where stochastic flow is invertible. See work by Kusuokaand Stroock in the uniformly elliptic case ([K-S], I, TaniguchiSympos. 1982).
(iii) The form of the Malliavin covariance allows polynomial (finite-type) rate of degeneracy near a hypersurface, coupled with lim-ited contact of the initial path with the hypersurface. cf. sodecase where degeneracies of infinite type are compatible with hy-pellipticity (Bell and Mohammed [B-M], Duke Math. Journal,1995).
3
Hypotheses (H).
(i) W : [0,∞) × Ω → Rm is standard m-dimensional Wiener process,defined on a complete filtered probability space (Ω,F , (Ft)t≥0, P ).
(ii) g : [0,∞)×Rd → Rd×m is a continuous map into the space of d×m
matrices, with bounded Frechet derivatives of all orders in thespace variables .
(iii) r is a positive real number, and η : [−r, 0] → Rd is a continuousinitial path.
(iv) H : [0,∞) × C → Rd is a globally bounded continuous map withall Frechet derivatives of H(t, η) wrt η globally bounded in (t, η) ∈R+ × C. Think of H(t, ξt) as a smooth Rd-valued functional inξ ∈ C([−r, t],Rd). Denote its Frechet derivative wrt ξ ∈ C([−r, t],Rd)by Hξ(t, ξ). Set
αt := sup‖Hξ(u, ξ)‖ : u ∈ [0, t], ξ ∈ C([−r, u],Rd) , t > 0 ,
andα∞ := sup‖Hξ(u, ξ)‖ : u ∈ [0,∞), ξ ∈ C([−r, u],Rd) ,
where ‖Hξ(u, ξ)‖ is the operator norm of the partial Frechet de-rivative Hξ(u, ξ) : C([−r, u],Rd) → Rd.
Theorem VI.1.
Assume Hypotheses (H) for the sfde (XIX). Suppose there exist positive
constants ρ, δ, an integer p ≥ 2 and a function
φ : [0,∞)×Rd → R satisfying the following conditions
(i)
g(t, x)g(t, x)∗ ≥ |φ(t, x)|pI, |φ(t, x)| < ρ
δI, |φ(t, x)| ≥ ρ(1)
4
for (t, x) ∈ [0,∞)×Rd.
(ii) φ(t, x) is C1 in t and C2 in x, with bounded first derivatives in (t, x) and
bounded second derivatives in x ∈ Rd.
(iii) There is a positive constant c such that
‖∇φ(t, x)‖ ≥ c > 0 (2)
for all (t, x) ∈ [0,∞)×Rd, with |φ(t, x)| ≤ ρ. In (2), ∇ denotes the gradient
operator with respect to the space variable x ∈ Rd.
(iv) There is a positive number δ0 such that δ0 < (3α∞)−1∧r and for every Borel
set J ⊆ [−r, 0] of Lebesgue measure δ0 the following holds
∫
J
φ(t + r, η(t))2 dt > 0. (3)
Define s0 ∈ [−r, 0] by
s0 := sups ∈ [−r, 0] :∫ s
−r
φ(u + r, η(u))2 du = 0.
Then for all t > s0 + r the solution x(t) of (XIX) is absolutely continuous
with respect to d-dimensional Lebesgue measure, and has a C∞ density.
Remark.
Condition (iv) of Theorem VI.1 may be replaced by the followingequivalent condition:(iv)′ The set s : s ∈ [−r, o], φ(s, η(s)) = 0 has Lebesgue measure less
than (3α∞)−1 ∧ r.
Theorem VI.2
In the sfde
dy(t) = H(t, yt) dt + F (t)dW (t), t > a
y(t) = x(t), a− r ≤ t ≤ a, a ≥ r
(XX)
5
suppose that F : [a,∞) → Rd×n and x : [0, a] → Rd are continuous. Assume that
H satisfies regularity hypotheses analogous to (H). For t > a let
α′t := sup‖Hξ(u, ξ)‖ : u ∈ [0, t], ξ ∈ C .
Suppose there exists δ∗ < 1/(3α′t) such that
∫ t
t−δ∗µ1(s) ds > 0, (4)
where µ1(s), s ≥ a, is the smallest eigenvalue of the non-negative definite matrix
F (s)F (s)∗. Then the solution y(t) of (XX) has an absolutely continuous distribution
with respect to d-dimensional Lebesgue measure and has a C∞ density.
In the special case when H(t, y) = h(t, y(t)) in equation (XX)for some Lipschitz function h : R+ × Rd → Rd, then y is a (time-inhomogeneous)diffusion process. In this case the proof of TheoremVI.2 gives the following pde result.
6
Theorem VI.3
For each t > 0, let A(t) = [aij(t)]di,j=1 denote a symmetric non-negative
definite d × d-matrix. Let µ2(t) be the smallest eigenvalue of A(t). Assume the
following:
(i) The map t 7−→ A(t) is continuous.
(ii) There exists T > 0 such that
∫ T
0
µ2(s) ds > 0. (5)
(iii) The functions bi, i = 1, . . . , d, c : R+ × Rd → R are bounded, jointly
continuous in (t, x) and have partial derivatives of all orders in x, all of
which are bounded in (t, x). Let T0 := supT > 0 :∫ T
0µ2(s) ds = 0, and
let Lt,x denote the differential operator
Lt,x :=12
d∑
i,j=1
aij(t)∂2
∂xi∂xj+
d∑
i=1
bi(t, x)∂
∂xi+ c(t, x). (6)
Then the parabolic equation∂u
∂t= Lt,xu has a fundamental solution Γ(t, x, y)
defined on (T0,∞) ×R2d, which is C1 in t and C∞ in (x, y). Furthermore,
if the coefficients aij(t), bi(t, x), c(t, x), i, j = 1, . . . , d, are C∞ in (t, x), and
limt→T0+
(t− T0) log∫ t
T0
µ2(s) ds
= 0, (7)
then∂
∂t− Lt,x is a hypoelliptic operator on (T0,∞) × Rd; (viz. if φ is a
distribution on (T0,∞)×Rd such that
(∂
∂t− Lt,x
)φ is C∞, then φ is also
C∞.)
The mean ellipticity hypothesis (5) is much weaker than classicalpointwise ellipticity .
Problem.
Can Theorem VI.3 be proved using existing pde’s techniques?
7
Proof of Theorem VI.1. (Outline)Objective is to get good probabilistic lower bounds on the Malli-
avin covariance matrix of the solution x(t) of (XIX). Do this using thefollowing steps:(cf.[B-M 2], Ann. Prob. 1995, “conditioning argument” using (XX)).Step 1.
We use piecewise linear approximations of W in (XIX) to com-pute the Malliavin covariance matrix C(T ) of x(T ) as
C(T ) =∫ T
0
Z(u)g(u, x(u− r)
)g(u, x(u− r)
)∗Z(u)∗ du,
where the (d × d)-matrix-valued process Z : [0, T ] × Ω → Rd×d satisfiesthe advanced anticipating Stratonovich integral equation
Z(t) = I +∫ T
T∧(t+r)
Z(u)Dg(u, x(u− r)
)(·) dW (u)
+∫ T
t
Z(u)[Hx(u, x)∗(·)′(t)]∗ du, 0 ≤ t ≤ T.
In the above integral equation, Hx(u, x) is the Frechet partial derivativeof the map
(u, x) → H(u, xu)
with respect to x ∈ C([−r, u],Rd
). If W 1,2 is the Cameron Martin sub-
space of C([−r, u],Rd
), let Hx(u, x)∗ denote the Hilbert-space adjoint of
the restrictionHx(u, x)|W 1,2 : W 1,2 → Rd.
We solve the above integral equation as follows.Start with the terminal condition Z(T ) = I. On the last delay
period [(T − r) ∨ 0, T ] define Z to be the unique solution of the linearintegral equation
Z(t) = I +∫ T
t
Z(u)[Hx(u, x)∗(·)′(t)]∗ du
8
for a.e. t ∈ ((T −r)∨0, T ). When T > r, use successive approximations tosolve the anticipating integral equation, treating the stochastic inte-gral as a predefined random forcing term. This gives a unique solutionof the integral equation by successive backward steps of length r. Thematrix Z(t) need not be invertible for small t. Compare Z(t) with theanalogous process for the diffusion case (sode). In this case Z(t) isinvertible for all t and anticipating integrals are not needed.
Step 2.Since Hx(u, x) is globally bounded in (u, x), then so is
[Hx(u, x)∗(·)]′(t) in (u, x, t) ([B-M], Ann. Prob. 1995, Lemma 3.3). Hencecan choose a deterministic time t0 < T sufficiently close to T such thatalmost surely Z(t) is invertible and ‖Z(t)−1‖ ≤ 2 for a.e. t ∈ (t0, T ].([B-M], Ann. Prob. 1995, Lemma 3.4).Step 3.
The above lower bound on ‖Z(t)‖ and the representation of C(T )imply that
det C(T ) ≥[14
∫ T
t0
g(u, x(u− r))2 du
]d
a.s.
whereg(u, v) := inf
|g(u, v)∗(e)| : e ∈ Rd, |e| = 1,
for all u ≥ 0, v ∈ Rd.Step 4.
Prove thePropagation Lemma:
Let −r < a < b < a + r. Then the statement
P
(∫ b
a
|φ(u + r, x(u))|2 du < ε
)= o(εk)
9
as ε → 0+ for every k ≥ 1,
implies that
P
(∫ b+r
a+r
|φ(u + r, x(u))|2 du < ε
)= o(εk)
as ε → 0+ for every k ≥ 1. (Proof uses Ito’s formula, the lower bound on‖∇φ‖, the polynomial degeneracy condition and the Kusuoka-Stroockε1/(18)-lemma!).Step 5.
By successively applying Step 4, we propagate the “limited con-tact” hypothesis on the initial path η in order to get the estimate:
P
(∫ T
t0
|φ(u, x(u− r))|2du < ε
)= o(εk)
as ε → 0+ for every k ≥ 1.Step 6.
Using the polynomial degeneracy hypothesis, Step 5, Jensen’sinequality, and Lemma 4.3 of ([B-M], Ann. Prob. 1995), we obtain
P
(∫ T
t0
g(u, x(u− r)
)2du < ε
)= o(εk)
as ε → 0+ for every k ≥ 1.Step 7.
Combining steps 3 and 6 gives
P(det C(T ) < ε
)= o(εk)
as ε → 0+ for every k ≥ 1. This implies that C(T )−1 exists a.s. anddetC(T )−1 ∈
∞⋂q=1
Lq(Ω,R). ¤
10
2. Back to Square One: Diffusions via SDDE’s
Objective is to prove the following existence theorem for classicaldiffusions using approximations by small delays: (Caratheodory)
Theorem VI.4.( Ito, Gihman-Skorohod,..)Let h : Rd → Rd, g : Rd → Rd×m be globally Lipschitz, and W m-
dimensional Brownian motion. Suppose x0 ∈ Rd. Then the sode
dx(t) = h(x(t)) dt + g(x(t)) dW (t), t > 0
x(0) = x0
has an adapted solution with continuous sample paths.
Proof.([B-M], Stochastics, 1989)For simplicity assume that h ≡ 0 and d = m = 1.Fix 0 < T < ∞. For each integer k ≥ 1, define
xk(t) = x0 +∫ t
0
g
(xk
(u− 1
k
))dW (u), t ≥ 0
xk(t) = x0, −1k≤ t ≤ 0
(∗)
Note that xk exists, is adapted and continuous.Step 1.
xk : [0,∞) → L2(Ω,R) is ( 12 )-Holder, with Holder constant indepen-
dent of k. (Exercise: To prove this observe first that by (∗) and thelinear growth property of g, there is a positive constant K independentof k and t ∈ [0, T ] such that
E sup0≤u≤t
(|xk(t)|2 + |g(xk(t)|2) ≤ K
for all k ≥ 1 and all t ∈ [0, T ]. Then E[xk(t) − xk(s)]2 ≤ K(t − s) for allt, s ≥ 0.)
11
Step 2.For each t ≥ 0, xk(t) converges to a limit x(t) in L2(Ω,Rd): Let L
be the Lipschitz constant for g.For l > k, we have
E[xl(t)− xk(t)]2 = E
∫ t
0
[g
(xl
(u− 1
l
))− g
(xk
(u− 1
k
))]dW (u)
2
≤ L2
∫ t
0
E
[xl
(u− 1
l
)− xk
(u− 1
l
)]2
du
≤ 2L2
∫ t
0
E
[xl
(u− 1
l
)− xk
(u− 1
l
)]2
du
+ 2L2
∫ t
0
E
[xk
(u− 1
l
)− xk
(u− 1
k
)]2
du
≤ 2L2
∫ t− 1l
− 1l
E[xl(u)− xk(u)]2 du + 2KL2t
(1k− 1
l
)
≤ 2L2
∫ t
0
E[xl(u)− xk(u)]2 du + 2t‖a‖∞(
1k− 1
l
)
by Step 1. Thus, by Gronwall’s lemma
E[xl(t)− xk(t)]2 ≤ 2TK
(1k− 1
l
)e2L2t
Thus convergence holds. Also
E[x(t)− xk(t)]2 ≤ 2TK
ke2L2t
Step 3.The process x satisfies original sode:
12
Simply take limits as k →∞ in both sides of (∗). ThenLHS → x(t) in L2. x is adapted, since each xk is. Also
E
∫ t
0
[g
(xk
(u− 1
k
))− g(x(u))
]dW (u)
2
≤ L2
∫ t
0
E
[xk(u− 1
k)− x(u)
]2
du
≤ 2L2
∫ t
0
E
[xk(u− 1
k)− xk(u)
]2
du + 2L2
∫ t
0
E[xk(u)− x(u)]2 du
≤ 2L2Kt
k+
2L2KT
k
∫ t
0
2e2L2u du
≤ 2KL2
k
[t +
12L2
(e2L2t − 1
)]
→ 0 as k →∞
ThusRHS → x0 +
∫ t
0
g(x(u)) dW (u)
i.e. x satisfies the sode.Since the Ito integral has an a.s. continuous modification, it
follows from Doob’s inequality that x has such a modification. ¤
3. Affine SFDE’s. A Simple Model of Population Growth
Recall simple population growth model (Lecture I):
dx(t) = −αx(t) + βx(t− r) dt + σdW (t), t > 0 (II)
for a large population x(t) withconstant birth rate = β > 0 (per capita);constant death rate = α > 0 (per capita);development period r = (9) > 0;
13
migration, white noise, variance σ.
To determine stability, growth rates of population, consider thegeneral affine system :
dx(t) =∫
[−r,0]
µ(ds)x(t + s)
dt + dQ(t), t > 0
x0 = η ∈ D := D([−r, 0],Rd).
(X)
D := space of all cadlag paths [−r, 0] → Rd with ‖ · ‖∞-norm.
14
Theorem VI.5.
The Lyapunov spectrum of (X) coincides with the set of all real parts βi : i ≥1 of the spectrum of the generator A of the homogeneous equation corresponding
to Q ≡ 0, together with possibly −∞.
We now consider the hyperbolic case when βi 6= 0 for all i ≥ 1.In this case, the following result ([M-S1],Theorem 20) establishes theexistence of a hyperbolic splitting along a unique stationary solutionof (X).
Theorem VI.6. ([M-S], 1990)Suppose that Q is cadlag and has stationary increments. Assume that the
characteristic equation
det(
λI −∫
[−r,0]
eλsµ(ds))
= 0
has no roots on the imaginary axis; i.e., the associated homogeneous equation (Q ≡0) has no zero Lyapunov exponents. Suppose also that
limt→±∞
1|t| log |Q(t)| < |Reλ| a.s.
for all characteristic roots λ. Then there is a unique D-valued random variable η∞such that the trajectory xη∞
t : t ≥ 0 of (X) is a D-valued stationary process.
The random variable η∞ is measurable with respect to the σ-algebra generated by
Q(t) : t ∈ R. Furthermore, let β1 > β2 > · · · , be an ordering of the Lyapunov
spectrum of (X) and suppose βm > 0, βm+1 < 0. Then there exists a decreasing
sequence of finite-codimensional subspaces Ei : i ≥ 1 of D such that
limt→∞
1t
log ‖xt(ω)‖∞ = βi, i ≥ 1
if x0(ω) ∈ η∞ + Ei−1\Ei, 1 ≤ i ≤ m, and
limt→∞
1t
log ‖xt(ω)‖∞ ≤ βm+1
15
if x0(ω) ∈ η∞ + Em.
Results on the existence of p-th moment Lyapunov exponentsappear in ([M-S], Stochastics, 1990). Under mild non-degeneracycondition on the stationary solution, one gets the existence of onlyone p-th moment exponent (= pβ1) which is independent of all random(possibly anticipating) initial conditions in D. This result is in agree-ment with the affine linear finite-dimensional non-delay case (r = 0)([AOP],[B],[AKO]).Problem.
Under what conditions on the parameters α, β does (II) have astationary solution?Interesting Fact:
The affine hereditary system (X) may be viewed as a finite-dimensional stochastic perturbation of the associated infinitely degener-ate deterministic homogeneous system (Q ≡ 0) with countably manyLyapunov exponents. However, these finite-dimensional perturba-tions provide noise that is generically rich enough to account for asingle p−th moment Lyapunov exponent in the affine stochastic sys-tem (X).
16
4. Random Delays
See [Mo], Pitman Books, 1984, pp. 167-186. Delays are allowedto be random (independently) of the noise and essentially bounded.Markov property fails, but get a measure-valued process with randomMarkov transition measures on state space.
5. Infinite Delays. Stationary Solutions
Pioneering work of Ito and Nisio, [I-N], J. Math. Kyoto Univer-sity, 1964, pp. 1-75. Results summarized in [Mo], 1984, pp. 230-233.
17
6. Summary
Main themes are:
SMOOTH HISTORY ⇐⇒ REGULARITY OF SFDE
FINITE HISTORY =⇒ LOCAL COMPACTNESS OF SEMI-FLOW =⇒ DISCRETE LYAPUNOV SPECTRUM =⇒ HYPER-BOLICITY AND STABLE MANIFOLDS
HELIX NOISE =⇒ NON-IVERTIBLE MULTIPLICATIVECOCYCLE ON HILBERT SPACE
DISCRETE FINITE HISTORY =⇒ SINGULAR SYSTEM =⇒NONEXISTENCE OF FLOW
18
STOCHASTIC DIFFERENTIAL SYSTEMS
WITH MEMORY
THEORY, EXAMPLES AND APPLICATIONS
Geilo, Norway : July 29-August 4, 1996
Salah-Eldin A. Mohammed
Southern Illinois University
Carbondale, IL 62901–4408 USA
Web page: http://salah.math.siu.edu
1
BIBLIOGRAPHIES (TENTATIVE)
Ergodic Theory and Lyapunov Exponents
[AKO] Arnold, L., Kliemann, W. and Oeljeklaus, E. Lyapunov expo-nents of linear stochastic systems, in Lyapunov Exponents, SpringerLecture Notes in Mathematics 1186 (1989), 85–125.
[AOP] Arnold, L. Oeljeklaus, E. and Pardoux, E., Almost sure andmoment stability for linear Ito equations, in Lyapunov Exponents,Springer Lecture Notes in Mathematics 1186 (ed. L. Arnoldand V. Wihstutz) (1986), 129–159.
[B] Baxendale, P.H., Moment stability and large deviations for lin-ear stochastic differential equations, in Ikeda, N. (ed.) Proceed-ings of the Taniguchi Symposium on Probabilistic Methods in MathematicalPhysics, Katata and Kyoto (1985), 31–54, Tokyo: Kinokuniya(1987).
[CJPS] Cinlar, E., Jacod, J., Protter, P. and Sharpe, M. Semimartin-gales and Markov processes, Z. Wahrsch. Verw. Gebiete 54(1980), 161–219.
[FS] Flandoli, F. and Schaumloffel, K.-U. Stochastic parabolic equa-tions in bounded domains: Random evolution operator andLyapunov exponents, Stochastics and Stochastic Reports 29,4 (1990), 461–485.
[M1] Mohammed, S.-E.A. Stochastic Functional Differential Equations, Re-search Notes in Mathematics 99, Pitman Advanced PublishingProgram, Boston-London-Melbourne (1984).
2
[M2] Mohammed, S.-E.A. Non-linear flows for linear stochastic delayequations, Stochastics 17, 3 (1986), 207–212.
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4
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[M3] Mohammed, S.-E.A., The Lyapunov spectrum and stable mani-folds for stochastic linear delay equations, Stochastics and StochasticReports 29 (1990), 89–131.
[M4] Mohammed, S.-E.A., Lyapunov exponents and stochastic flowsof linear and affine hereditary systems, (1992) (Survey article),in Diffusion Processes and Related Problems in Analysis, Volume II , editedby Pinsky, M., and Wihstutz, V., Birkhauser (1992), 141–169.
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5
[S] Scheutzow, M.K.R. Stationary and Periodic Stochastic Differential Sys-tems: A study of Qualitative Changes with Respect to the Noise Level andAsymptotics, Habiltationsschrift, University of Kaiserslautern, W.Germany (1988).
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[B-M.3] D.R. Bell and S.-E. A. Mohammed, Operateurs paraboliqueshypoelliptiques avec degenerescences exponentielles, C.R. Acad.Sci. Paris, t. 317, Serie I, (1993), 1059-1064.
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6
[M.2] S.-E.A. Mohammed, Non-linear flows for linear stochastic delayequations, Stochastics, Vol.17, No. 3 (1986), 207-212.
[S] D.W. Stroock, The Malliavin calculus, a functional analytic ap-proach, J. Funct. Anal. 44 (1981), 212–257.
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[B-M.2] D.R. Bell and S.-E. A. Mohammed, An extension of Hormander’stheorem for infinitely degenerate parabolic operators, preprint.
[B-M.3] D.R. Bell and S.-E. A. Mohammed, Operateurs paraboliqueshypoelliptiques avec degenerescences exponentielles, C.R. Acad.Sci. Paris, t. 317, Serie I, (1993), 1059-1064.
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7
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Stochastic delay equations
[1] 1 379 172 Bell, Denis R.; Mohammed, Salah-Eldin A. Smoothdensities for degenerate stochastic delay equations with hereditarydrift. Ann. Probab. 23 (1995), no. 4, 1875–1894.60H07 (60H1060H20 60H30)
[2] 92k:60124 Bell, Denis R.; Mohammed, Salah Eldin A. TheMalliavin calculus and stochastic delay equations. J. Funct. Anal. 99(1991), no. 1, 75–99. (Reviewer: David Nualart) 60H10 (60H07)
[3] 92a:60148 Mohammed, Salah Eldin A.; Scheutzow, MichaelK. R. Lyapunov exponents and stationary solutions for affine stochas-tic delay equations. Stochastics Stochastics Rep. 29 (1990), no. 2,259–283. (Reviewer: Wolfgang Kliemann) 60H20 (34D08 93D05)
[4] 91c:49045 Flandoli, Franco Solution and control of a bilinearstochastic delay equation. SIAM J. Control Optim. 28 (1990), no. 4,936–949. (Reviewer: Negash G. Medhin) 49K45 (60H20 93C20)
[5] 90b:60068 Mohammed, S. E. A. Unstable invariant distri-butions for a class of stochastic delay equations. Proc. EdinburghMath. Soc. (2) 31 (1988), no. 1, 1–23. (Reviewer: Stanislaw Wpolhkedrychowicz) 60H10 (34F05 34K05)
[6] 89b:60154 Tudor, C.; Tudor, M. On approximation of solu-tions for stochastic delay equations. Stud. Cerc. Mat. 39 (1987), no.3, 265–274. (Reviewer: Seppo Heikkild) 60H20 (34F05 34K05)
[7] 88j:60082 Chang, Mou Hsiung Discrete approximation ofnonlinear filtering for stochastic delay equations. Stochastic Anal.Appl. 5 (1987), no. 3, 267–298. (Reviewer: Maurizio Pratelli) 60G35(60H10 93E11 93E25)
[8] 87j:60097 Mohammed, S.; Scheutzow, M.; von Weizsdcker,H. Hyperbolic state space decomposition for a linear stochastic de-lay equation. SIAM J. Control Optim. 24 (1986), no. 3, 543–551.(Reviewer: Ludwig Arnold) 60H99 (34K20)
[9] 85h:60087 Scheutzow, M. Qualitative behaviour of stochasticdelay equations with a bounded memory. Stochastics 12 (1984), no.1, 41–80. (Reviewer: Toshio Yamada) 60H10
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[10] 80a:60083 Chojnowska-Michalik, Anna Representation the-orem for general stochastic delay equations. Bull. Acad. Polon. Sci.Sir. Sci. Math. Astronom. Phys. 26 (1978), no. 7, 635–642. (Re-viewer: I. M. Stoyanov) 60H20 (34G99 34K99)
Stochastic functional differential equations
[1] 1 373 727 Mohammed, Salah-Eldin A.; Scheutzow, MichaelK. R. Lyapunov exponents of linear stochastic functional differentialequations driven by semimartingales. I. The multiplicative ergodictheory. Ann. Inst. H. Poincari Probab. Statist. 32 (1996), no. 1,69–105.60H99
[2] 1 355 171 Yasinskii, V. K.; Yurchenko, I. V. Asymptoticstability in the mean square of the trivial solution of systems of sto-chastic functional-differential equations with discontinuous trajecto-ries. (Ukrainian) Dopov./Dokl. Akad. Nauk Ukraoni 1994, no. 11,24–28.34K50 (34K20 60H99)
[3] 1 318 857 Turo, Jan Successive approximations of solutions tostochastic functional-differential equations. Period. Math. Hungar.30 (1995), no. 1, 87–96.60H10 (34Kxx)
[4] 1 308 293 Kolmanovskii, V.; Shaikhet, L. New results in sta-bility theory for stochastic functional-differential equations (SFDEs)and their applications. Proceedings of Dynamic Systems and Appli-cations, Vol. 1 (Atlanta, GA, 1993), 167–171, Dynamic, Atlanta, GA,1994. 60Hxx (34Kxx 93E15)
[5] 96f:60110 Wang, Xiang Dong; Zhang, Dao Fa Existence ofsolutions for stochastic functional-differential equations with contin-uous coefficients. (Chinese) J. Math. Res. Exposition 15 (1995), no.2, 267–270. (Reviewer: Binggen Zhang) 60H99
[6] 95i:34153 Kadiev, R. I. The admissibility of pairs of spaceswith respect to some of the variables for linear stochastic functional-differential equations. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat.1994, no. 5, 13–22. (Reviewer: Binggen Zhang) 34K50 (34K20 60H99)
[7] 94i:60069 Xu, Jia Gu Weak and strong solutions of gener-alized Itt’s stochastic functional-differential equations. Acta Math.
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Sinica (N.S.) 9 (1993), no. 3, 265–277. (Reviewer: Vivek S. Borkar)60H10 (34K50)
[8] 93j:60091 Kadiev, R. I. Stability with respect to some of thevariables of linear stochastic functional-differential equations. (Rus-sian) Functional-differential equations (Russian), 140–148, Perm. Po-litekh. Inst., Perm, 1991. 60H99 (34K20 34K50)
[9] 93i:60119 Bainov, D. D.; Kolmanovskii, V. B. Periodic solu-tions of stochastic functional-differential equations. Math. J. ToyamaUniv. 14 (1991), 1–39.60H99 (34K15 34K20 34K50)
[10] 93i:34147 Kadiev, R. I.; Ponosov, A. V. Stability of lin-ear stochastic functional-differential equations with constantly actingperturbations. (Russian) Differentsialnye Uravneniya 28 (1992), no.2, 198–207, 364; translation in Differential Equations 28 (1992), no.2, 173–179 34K50 (34K20)
[11] 93h:60102 Nechaeva, I. G.; Khusainov, D. Ya. Obtain-ing estimates for the stability of solutions of stochastic functional-differential equations. (Russian) Differentsialnye Uravneniya 28 (1992),no. 3, 405–414, 547; translation in Differential Equations 28 (1992),no. 3, 338–346 60H99 (34K20 34K50)
[12] 93g:49007 Govindan, T. E.; Joshi, M. C. Stability and opti-mal control of stochastic functional-differential equations with mem-ory. Numer. Funct. Anal. Optim. 13 (1992), no. 3-4, 249–265.49J25(34K35 49K45 93E15 93E20)
[13] 92e:60129 Nechaeva, I. G.; Khusainov, D. Ya. Exponen-tial estimates for solutions of linear stochastic functional-differentialequations. (Russian) Ukrain. Mat. Zh. 42 (1990), no. 10, 1338–1343; translation in Ukrainian Math. J. 42 (1990), no. 10, 1189–1193(1991) (Reviewer: Binggen Zhang) 60H99
[14] 91j:60112 Zhang, Dao Fa; Wang, Xiang Dong Some noteson stochastic functional-differential equations. (Chinese) Qufu ShifanDaxue Xuebao Ziran Kexue Ban 16 (1990), no. 3, 24–27.60H99
[15] 91i:60169 Tsarkov, E. F.; Yasinskii, V. K. Nonlinear sto-chastic functional-differential equations. (Russian) Akad. Nauk Ukrain.SSR Inst. Mat. Preprint 1989, no. 32, 27–46. (Reviewer: BinggenZhang) 60H99 (47H99)
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[16] 91g:60075 Marcus, Moshe; Mizel, Victor J. Stochastic func-tional differential equations modelling materials with selective recall.Stochastics 25 (1988), no. 4, 195–232. (Reviewer: S. E. A. Mo-hammed) 60H99 (34F05 34K99 35R60 93E03)
[17] 91d:60150 Rodkina, A. E. Stochastic functional-differentialequations with respect to a semimartingale. (Russian) Differentsial-nye Uravneniya 25 (1989), no. 10, 1716–1721, 1835; translation inDifferential Equations 25 (1989), no. 10, 1195–1120 (1990) (Reviewer:V. G. Kolomiets) 60H99 (34F05 34K99 60G44)
[18] 90k:60113 Rodkina, A. E. A proof of the solvability of non-linear stochastic functional-differential equations. (Russian) Globalanalysis and nonlinear equations (Russian), 127–133, 174, Novoe Global.Anal., Voronezh. Gos. Univ., Voronezh, 1988. 60H10 (34F05 34K99)
[19] 90f:60116 Kadiev, R. I. Solvability of the Cauchy problemfor stochastic functional-differential equations. (Russian) Functional-differential equations (Russian), 75–83, vi, Perm. Politekh. Inst.,Perm, 1987. 60H99
[20] 89m:60142 Rodkina, A. E. The averaging principle for sto-chastic functional-differential equations. (Russian) DifferentsialnyeUravneniya 24 (1988), no. 9, 1543–1551, 1653; translation in Differ-ential Equations 24 (1988), no. 9, 1014–1021 (Reviewer: ConstantinTudor) 60H20 (34C29)
[21] 89e:93215 Sverdan, M. L.; Yasinskaya, L. I. Exponential sta-bility of stochastic functional-differential equations under constantlyacting random perturbations. (Russian) Dokl. Akad. Nauk Ukrain.SSR Ser. A 1987, no. 4, 18–21.93E15 (34F05 60H99)
[22] 89b:60153 Tudor, C. On stochastic functional-differentialequations. Differential equations and applications, I, II (Russian)(Ruse, 1985), 971–974, ‘Angel Kancev’ Tech. Univ., Ruse, 1987. (Re-viewer: Seppo Heikkild) 60H20 (34K05)
[23] 88m:60173 Tudor, Constantin On stochastic functional-differentialequations with unbounded delay. SIAM J. Math. Anal. 18 (1987),no. 6, 1716–1725. (Reviewer: B. G. Pachpatte) 60H99
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[24] 87a:34066 Sverdan, M. L.; Yasinskaya, L. I. Exponentialstability in the mean square of solutions of stochastic functional-differential equations in the presence of Poisson perturbations. (Rus-sian) Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 1985, no. 41,20–26.34F05 (34K20)
[25] 86j:60151 Mohammed, S. E. A. Stochastic functional differ-ential equations. Research Notes in Mathematics, 99. Pitman (Ad-vanced Publishing Program), Boston, Mass.-London, 1984. vi+245pp. ISBN: 0-273-08593-X (Reviewer: M. Mitivier) 60H99 (34F05)
[26] 86f:60079 Chang, Mou Hsiung On Razumikhin-type stabil-ity conditions for stochastic functional differential equations. Math.Modelling 5 (1984), no. 5, 299–307. (Reviewer: S. M. Khrisanov)60H25 (34F05)
[27] 84g:34119 Mohammed, S. E. A. The infinitesimal generatorof a stochastic functional-differential equation. Ordinary and partialdifferential equations (Dundee, 1982), pp. 529–538, Lecture Notesin Math., 964, Springer, Berlin-New York, 1982. (Reviewer: G. S.Ladde) 34K05 (34F05 60H25)
[28] 83a:60096 Sasagawa, I. Existence of stationary and periodicsolutions of stochastic functional-differential equations. Proceedingsof the VIIIth International Conference on Nonlinear Oscillations, Vol.II (Prague, 1978), pp. 603–608, Academia, Prague, 1979. 60H10(34F05)
[29] 82k:60128 Jasinskaja, L. I. Mean square stability of thetrivial solution of linear stochastic functional-differential equationswith variable coefficients. (Russian) Ukrain. Mat. Zh. 33 (1981), no.4, 482–488. (Reviewer: D. Bobrowski) 60H10 (93E15)
[30] 82g:93049 Jasinskii , V. K. Stability of solutions of linearstochastic functional-differential equations with variable coefficients.(Russian) Analytic methods for investigating nonlinear oscillations(Russian), pp. 169–177, 185, Akad. Nauk Ukrain. SSR, Inst. Mat.,Kiev, 1980. 93E15
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[31] 82b:60076 Jasinskii , V. K. Stability of the almost surelytrivial solution of stochastic functional-differential equations. (Rus-sian) Asymptotic methods of nonlinear mechanics (Russian), pp. 176–184, 200, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979. (Re-viewer: G. N. Milshtei n) 60H25 (34F05)
[32] 82a:60087 Jasinskaja, L. I.; Jasinskii , V. K. Asymptoticmean-square stability of the trivial solution of stochastic functional-differential equations. (Russian) Ukrain. Mat. Zh. 32 (1980), no. 1,89–98, 143. (Reviewer: D. Bobrowski) 60H10 (34K20)
[33] 80m:49023 Chang, Mou Hsiung Optimal control for stochas-tic functional differential equations. Bull. Inst. Math. Acad. Sinica7 (1979), no. 1, 21–37. (Reviewer: Virginia M. Warfield) 49B60(34F05)
[34] 58 #13349 Carkov, E. F. Stability in the first approximationof the trivial solution of stochastic functional-differential equations.(Russian) Mat. Zametki 23 (1978), no. 5, 733–738. (Reviewer: Mou-Hsiung Chang) 60H10 (34K20)
[35] 58 #9692 Carkov, E. F. Exponential p-stability of the triv-ial solution of stochastic functional-differential equations. (Russian)Teor. Verojatnost. i Primenen. 23 (1978), no. 2, 445–448. (Reviewer:I. M. Stojanov) 93E15 (60H20)
[36] 58 #6572 Carkov, E. F. Asymptotic stability of the solutionsof stochastic functional-differential equations. (Russian) Approximatemethods for the study of nonlinear systems (Russian), pp. 221–232.Izdanie Inst. Mat. Akad. Nauk SSR, Kiev, 1976. 34F05 (60H10)
[37] 58 #3046 Carkov, E. F. The structure of the spectrumof the generating operator of a semigroup of covariance operators ofthe solution of stochastic functional-differential equations. (Russian)Latviisk. Mat. Ezegodnik Vyp. 21 (1977), 99–107, 236. (Reviewer:N. N. Vahanija (Vakhania)) 60H99
[38] 57 #10827 Chang, Mou Hsiung On existence and stabilityresults of a class of stochastic functional differential equations. Bull.Inst. Math. Acad. Sinica 4 (1976), no. 2, 209–228. (Reviewer: KunioNishioka) 60H99 (34K05 34K20)
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[39] 55 #4381 Carkov, E. F. Mean-square asymptotic exponen-tial stability of the trivial solution of stochastic functional-differentialequations. (Russian) Teor. Verojatnost. i Primenen. 21 (1976), no.4, 871–875. (Reviewer: I. M. Stojanov) 60H99 (93E15)
[40] 50 #11443 Chang, M. H.; Ladde, G.; Liu, P. T. Stabilityof stochastic functional differential equations. J. Mathematical Phys.15 (1974), 1474–1478. (Reviewer: E. F. Carkov) 60H10 (34KXX)
[41] 49 #8105 Ladde, G. S. Differential inequalities and stochas-tic functional differential equations. J. Mathematical Phys. 15 (1974),738–743. (Reviewer: A. T. Bharucha-Reid) 60H10 (34F05)
[42] 49 #6352 Bikis, M. A.; Sadovjak, A. M.; Carkov, E. F.Continuity with respect to a parameter of the solutions of stochas-tic functional-differential equations. (Russian) Latvian mathematicalyearbook, 12 (Russian), pp. 39–49. Izdat. ”Zinatne”, Riga, 1973.(Reviewer: G. S. Ladde) 60H10 (34F05)
[43] 1 222 665 Ponosov, A. V. On the theory of reducible stochas-tic functional-differential equations with respect to a semimartingale.(Russian) Functional-differential equations (Russian), 111–120, Perm.Politekh. Inst., Perm, 1990. 60H99 (34K50)
[44] 1 221 339 Sergeev, V. A. The property of exponential p-stability of the trivial solution of a periodic stochastic functional-differential equation. (Russian) Boundary value problems (Russian),102–106, Perm. Politekh. Inst., Perm, 1991. 34K50 (34K20 60H99)
[45] 1 219 110 Ponosov, A. V. Canonical reducibility of stochas-tic functional-differential equations. (Russian) Boundary value prob-lems (Russian), 75–79, Perm. Politekh. Inst., Perm, 1989. 34K50
[46] 1 188 476 Yasinsky, V. K. On strong solutions of stochasticfunctional-differential equations with infinite aftereffect. Proceedingsof the Latvian Probability Seminar, Vol. 1 (Riga, 1991), 189–215,Riga Tech. Univ., Riga, 1992. 60H99
[47] 1 188 474 Tsarkov, Je.; Yanson, V. A. Investigation of sta-bility of stochastic functional-differential equations with periodic co-efficients. Proceedings of the Latvian Probability Seminar, Vol. 1(Riga, 1991), 154–174, Riga Tech. Univ., Riga, 1992. 60H99
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