Some linear SPDEs with fractional noiseaix1.uottawa.ca/~rbalan/Banff-slides.pdfOutline 1 Linear SPDEs with fractional noise 2 A parabolic equation 3 A hyperbolic equation Raluca Balan

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Some linear SPDEs with fractional noise

Raluca Balan

University of Ottawa

Workshop on Stochastic Analysis and SPDEsApril 2-6, 2012

Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 1 / 21

Outline

1 Linear SPDEs with fractional noise

2 A parabolic equation

3 A hyperbolic equation


Linear SPDEs with fractional noise

Linear SPDEs

Let L be a second order pseudo-differential operator in (t , x). Assumethat the fundamental solution G of Lu = 0 exists. Consider the linearSPDE: {

Lu(t , x) = W (t , x), t > 0, x ∈ Rd

zero initial conditions(1)

Space-time white noise

{W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:

E [W (ϕ)W (ψ)] =

∫ ∞0

∫Rdϕ(t , x)ψ(t , x)dxdt

Isometry The map ϕ 7→W (ϕ) is extended to L2(R+ × Rd )

W (ϕ) :=

∫ ∞0

∫Rdϕ(t , x)W (dt ,dx) (stochastic integral)



Linear SPDEs

Let L be a second order pseudo-differential operator in (t , x). Assumethat the fundamental solution G of Lu = 0 exists. Consider the linearSPDE: {

Lu(t , x) = W (t , x), t > 0, x ∈ Rd

zero initial conditions(1)

Space-time white noise

{W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:

E [W (ϕ)W (ψ)] =

∫ ∞0

∫Rdϕ(t , x)ψ(t , x)dxdt

Isometry The map ϕ 7→W (ϕ) is extended to L2(R+ × Rd )

W (ϕ) :=

∫ ∞0

∫Rdϕ(t , x)W (dt ,dx) (stochastic integral)



Definition (Walsh, 1986)

The process {u(t , x); t ≥ 0, x ∈ Rd} defined by

u(t , x) =

∫ t

0

∫Rd

G(t − s, x − y)W (ds,dy) (2)

is a random field solution of (1), provided that the stochastic integralin the RHS of (2) is well-defined.

Remark If W is a space-time white noise, (1) has a random-fieldsolution iff G(t − ·, x − ·) ∈ L2(R+ × Rd ).

Example Let L = ∂∂t −

12∆. Then G(t , x) = 1

(2πt)d/2 exp{− |x |

2

2t

}and

∫ t

0

∫Rd

G2(t − s, x − y)dyds <∞⇐⇒ d = 1





u(t , x) =

∫ t

0

∫Rd

G(t − s, x − y)W (ds,dy) (2)




12∆. Then G(t , x) = 1

(2πt)d/2 exp{− |x |

2

2t

}and

∫ t

0

∫Rd

G2(t − s, x − y)dyds <∞⇐⇒ d = 1





u(t , x) =

∫ t

0

∫Rd

G(t − s, x − y)W (ds,dy) (2)




12∆. Then G(t , x) = 1

(2πt)d/2 exp{− |x |

2

2t

}and

∫ t

0

∫Rd

G2(t − s, x − y)dyds <∞⇐⇒ d = 1



Colored Noise (Dalang and Frangos, 1998)

W = {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:

E [W (ϕ)W (ψ)] =

∫ ∞0

∫Rd

∫Rdϕ(t , x)ψ(t , y)f (x − y)dxdydt =: J(ϕ,ψ)

J is non-negative definite iff ∃µ tempered measure with f = Fµ∫Rd

∫Rdϕ(x)ψ(y)f (x−y)dxdy =

∫RdFϕ(ξ)Fψ(ξ)µ(dξ), ∀ϕ,ψ ∈ S(Rd )

Examples

1. Riesz kernel µ(dξ) = |ξ|−αdξ, f (x) = cα,d |x |−(d−α), 0 < α < d2. Bessel kernel µ(dξ) = (1 + |ξ|2)−α/2dξ, α > 0

Isometry

ϕ 7→W (ϕ) is extended to a Hilbert space P, which may containdistributions in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 5 / 21




E [W (ϕ)W (ψ)] =

∫ ∞0

∫Rd





Examples


Isometry





E [W (ϕ)W (ψ)] =

∫ ∞0

∫Rd





Examples


Isometry



Theorem 1 (Dalang, 1999)

Assume G(t , ·) ∈ S ′(Rd ) has rapid decrease (and satisfies someregularity conditions). Then G(t − ·, x − ·) belongs to the space P iff∫ t

0

∫Rd|FG(s, ·)(ξ)|2µ(dξ)ds <∞ (3)

Example

Let L = ∂∂t − L, where L is the L2-generator of a Lévy process (Xt )t≥0

with values in Rd . Let Ψ(ξ) be the characteristic exponent of (Xt )t≥0:

E(e−iξ·Xt ) = e−tΨ(ξ)

Assume that Xt has density pt . Then G(t , x) = pt (−x) and henceFG(t , ·)(ξ) = e−tΨ(ξ). Condition (3) holds iff

Υ(1) :=

∫Rd

11 + 2ReΨ(ξ)

µ(dξ) <∞Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 6 / 21


Theorem 1 (Dalang, 1999)

Assume G(t , ·) ∈ S ′(Rd ) has rapid decrease (and satisfies someregularity conditions). Then G(t − ·, x − ·) belongs to the space P iff∫ t

0

∫Rd|FG(s, ·)(ξ)|2µ(dξ)ds <∞ (3)

Example

Let L = ∂∂t − L, where L is the L2-generator of a Lévy process (Xt )t≥0

with values in Rd . Let Ψ(ξ) be the characteristic exponent of (Xt )t≥0:

E(e−iξ·Xt ) = e−tΨ(ξ)

Assume that Xt has density pt . Then G(t , x) = pt (−x) and henceFG(t , ·)(ξ) = e−tΨ(ξ). Condition (3) holds iff

Υ(1) :=

∫Rd

11 + 2ReΨ(ξ)

µ(dξ) <∞Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 6 / 21


Fractional Brownian Motion (FBM)

Let H ∈ (0,1). FBM is a centered Gaussian process (Bt )t≥0 with

E(BtBs) =12

(t2H + s2H − |t − s|2H) =: RH(t , s)

Important Remark

If H > 1/2, then RH(t , s) = αH∫ t

0

∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)

B. and Tudor (2008)

Assume H > 1/2. Let {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} be a centeredGaussian process with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H

〈ϕ,ψ〉H = αH

∫R2

+

∫R2d

ϕ(u, x)ψ(v , y)|u − v |2H−2f (x − y)dxdydudv

Isometry: ϕ 7→W (ϕ) is extended to a Hilbert space H which maycontain distributions in t , and in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 7 / 21




E(BtBs) =12

(t2H + s2H − |t − s|2H) =: RH(t , s)

Important Remark


0

∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)

B. and Tudor (2008)


〈ϕ,ψ〉H = αH

∫R2

+

∫R2d






E(BtBs) =12

(t2H + s2H − |t − s|2H) =: RH(t , s)

Important Remark


0

∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)

B. and Tudor (2008)


〈ϕ,ψ〉H = αH

∫R2

+

∫R2d




Theorem 2 (B. and Tudor, 2010)

Assume G(t , ·) ∈ S ′(Rd ) and FG(t , ·) is a function (satisfying someregularity conditions). Then G(t − ·, x − ·) belongs to the space H iff

It =

∫Rd

∫ t

0

∫ t

0|r − s|2H−2FG(r , ·)(ξ)FG(s, ·)(ξ)drdsµ(dξ) <∞ (4)

Examples

1. L = ∂∂t + (−∆)β/2, β > 0. Condition (4) holds iff∫

Rd

(1

1 + |ξ|β

)2H

µ(dξ) <∞

2. L = ∂2

∂t2 + (−∆)β/2, β > 0. Condition (4) holds iff∫Rd

(1

1 + |ξ|β

)H+1/2

µ(dξ) <∞

3. L = ∂∂t − L, where L is the generator of a Lévy process with values

in Rd and characteristic exponent Ψ

(4)⇐⇒∫Rd

(1

1 + ReΨ(ξ)

)2H

µ(dξ) <∞



Theorem 2 (B. and Tudor, 2010)

Assume G(t , ·) ∈ S ′(Rd ) and FG(t , ·) is a function (satisfying someregularity conditions). Then G(t − ·, x − ·) belongs to the space H iff

It =

∫Rd

∫ t

0

∫ t

0|r − s|2H−2FG(r , ·)(ξ)FG(s, ·)(ξ)drdsµ(dξ) <∞ (4)

Examples

1. L = ∂∂t + (−∆)β/2, β > 0. Condition (4) holds iff∫

Rd

(1

1 + |ξ|β

)2H

µ(dξ) <∞

2. L = ∂2

∂t2 + (−∆)β/2, β > 0. Condition (4) holds iff∫Rd

(1

1 + |ξ|β

)H+1/2

µ(dξ) <∞

3. L = ∂∂t − L, where L is the generator of a Lévy process with values

in Rd and characteristic exponent Ψ

(4)⇐⇒∫Rd

(1

1 + ReΨ(ξ)

)2H

µ(dξ) <∞


A parabolic equation


The equation

W is a Gaussian noise with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H. We consider

∂u∂t

(t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (5)

with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). We assumethat Xt has density pt .

The random-field solutionBy Theorem 2, (5) has a random-field solution iff

∫Rd Nt (ξ)µ(dξ) <∞,

Nt (ξ) = αH

∫ t

0

∫ t

0|r − s|2H−2e−rΨ(ξ)e−sΨ(ξ)drds




The equation

W is a Gaussian noise with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H. We consider

∂u∂t

(t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (5)

with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). We assumethat Xt has density pt .

The random-field solutionBy Theorem 2, (5) has a random-field solution iff

∫Rd Nt (ξ)µ(dξ) <∞,

Nt (ξ) = αH

∫ t

0

∫ t

0|r − s|2H−2e−rΨ(ξ)e−sΨ(ξ)drds



Theorem 3 (B. 2011)

For any t > 0, ξ ∈ Rd ,

c(1)H

(1

1/t + ReΨ(ξ)

)2H

≤ Nt (ξ) ≤ c(2)H

(1

1/t + ReΨ(ξ)

)2H

For the lower bound, we assume that there exists a constant K > 0such that

|ImΨ(ξ)| ≤ K ReΨ(ξ), for all ξ ∈ Rd . (6)

CorollaryIf (6) holds, then equation (5) has a random field solution iff

ΥH(1) :=

∫Rd

(1

1 + 2ReΨ(ξ)

)2H

µ(dξ) <∞ (7)



Theorem 3 (B. 2011)


c(1)H

(1

1/t + ReΨ(ξ)

)2H

≤ Nt (ξ) ≤ c(2)H

(1

1/t + ReΨ(ξ)

)2H

For the lower bound, we assume that there exists a constant K > 0such that

|ImΨ(ξ)| ≤ K ReΨ(ξ), for all ξ ∈ Rd . (6)

CorollaryIf (6) holds, then equation (5) has a random field solution iff

ΥH(1) :=

∫Rd

(1

1 + 2ReΨ(ξ)

)2H

µ(dξ) <∞ (7)



Example

(Xt )t≥0 is a symmetric β-stable Lévy process with values in Rd , forsome β ∈ (0,2]. In this case,

L = −(−∆)β/2 and Ψ(ξ) = |ξ|β

a) Assume that µ(dξ) = |ξ|−αdξ with α ∈ (0,d). Then (6) holds iff

2Hβ > d − α

b) Assume that µ(dξ) =∏d

i=1(cHi |ξi |−(2Hi−1))dξ with Hi ∈ (1/2,1).Then (6) holds iff

2Hβ > d −d∑

i=1

(2Hi − 1)



Potential Theory

Let X t = Xt − Xt , where (Xt )t≥0 is an independent copy of (Xt )t≥0.

(P tφ)(x) =

∫Rdφ(y)pt (x − y)dy , t ≥ 0 semigroup of (X t )t≥0

(Rαφ)(x) =

∫ ∞0

e−αs(Psφ)(x)ds, α > 0 resolvent of (X t )t≥0

pt = pt ∗ pt is the density of X t , where pt (x) = pt (−x).

Maximum Principle (Foondun and Khoshnevisan, 2010)

If µ is a tempered measure in Rd which has a density g, and f = Fµ inS ′(Rd ), then for any α > 0

(Rαf )(0) = supx∈Rd

(Rαf )(x) = Υ(α) :=

∫Rd

1α + 2ReΨ(ξ)

µ(dξ)



Potential Theory

Let X t = Xt − Xt , where (Xt )t≥0 is an independent copy of (Xt )t≥0.

(P tφ)(x) =

∫Rdφ(y)pt (x − y)dy , t ≥ 0 semigroup of (X t )t≥0

(Rαφ)(x) =

∫ ∞0

e−αs(Psφ)(x)ds, α > 0 resolvent of (X t )t≥0

pt = pt ∗ pt is the density of X t , where pt (x) = pt (−x).

Maximum Principle (Foondun and Khoshnevisan, 2010)

If µ is a tempered measure in Rd which has a density g, and f = Fµ inS ′(Rd ), then for any α > 0

(Rαf )(0) = supx∈Rd

(Rαf )(x) = Υ(α) :=

∫Rd

1α + 2ReΨ(ξ)

µ(dξ)



Fractional Analogues

(Rα,Hφ)(x) = αH

∫ ∞0

∫ ∞0|r − s|2H−2e−α(r+s)(Pr+sφ)(x)drds

Υ∗H(α) = αH

∫Rd

∫ ∞0

∫ ∞0|r − s|2H−2e−(α+2ReΨ(ξ))(r+s)drdsµ(dξ)

Theorem 4 (B., 2011)

Let H > 1/2. If µ is a tempered measure in Rd which has a density g,and f = Fµ in S ′(Rd ), then for any α > 0

(Rα,H f )(0) = supx∈Rd

(Rα,H f )(x) = Υ∗H(α)

Remark: CH,αΥH(α) ≤ Υ∗H(α) ≤ CHΥH(α) where

ΥH(α) =

∫Rd

(1

α + 2ReΨ(ξ)

)2H

µ(dξ)



Fractional Analogues

(Rα,Hφ)(x) = αH

∫ ∞0

∫ ∞0|r − s|2H−2e−α(r+s)(Pr+sφ)(x)drds

Υ∗H(α) = αH

∫Rd

∫ ∞0

∫ ∞0|r − s|2H−2e−(α+2ReΨ(ξ))(r+s)drdsµ(dξ)

Theorem 4 (B., 2011)

Let H > 1/2. If µ is a tempered measure in Rd which has a density g,and f = Fµ in S ′(Rd ), then for any α > 0

(Rα,H f )(0) = supx∈Rd

(Rα,H f )(x) = Υ∗H(α)

Remark: CH,αΥH(α) ≤ Υ∗H(α) ≤ CHΥH(α) where

ΥH(α) =

∫Rd

(1

α + 2ReΨ(ξ)

)2H

µ(dξ)



Intersection local time

(X(1)t )t≥0 and (X

(2)t )t≥0 are two independent copies of (X t )t≥0

For any nonnegative measurable function φ, we define

Lt ,H(φ) = α

∫ t

0

∫ t

0|r − s|2H−2φ(X

(1)r − X

(2)s )drds

Lt ,H(φ) may be infinite.

Since X(1)r − X

(2)s

d= X r+s,

E [f (X(1)r − X

(2)s )] = E [f (X r+s)] = (Pr+sf )(0)

E [Lt ,H(f )] = αH

∫ t

0

∫ t

0|r − s|2H−2(Pr+sf )(0)drds



Intersection local time

(X(1)t )t≥0 and (X

(2)t )t≥0 are two independent copies of (X t )t≥0

For any nonnegative measurable function φ, we define

Lt ,H(φ) = α

∫ t

0

∫ t

0|r − s|2H−2φ(X

(1)r − X

(2)s )drds

Lt ,H(φ) may be infinite.

Since X(1)r − X

(2)s

d= X r+s,

E [f (X(1)r − X

(2)s )] = E [f (X r+s)] = (Pr+sf )(0)

E [Lt ,H(f )] = αH

∫ t

0

∫ t

0|r − s|2H−2(Pr+sf )(0)drds



Lemma (B. 2011)

If (Rα,H f )(0) <∞ for any α > 0, then Lt ,H(f ) <∞ for all t > 0 a.s.Moreover,

lim supt→∞

1t

log Lt ,H(f ) ≤ 0 a.s.

Example

(Xt )t≥0 is a symmetric β-stable process, β ∈ (0,2].f (x) = cα,d |x |−(d−α) with α ∈ (0,d).Since (Xt )t≥0 is self-similar with exponent 1/β,

X r+sd= (r + s)1/βX 1

E [f (X r+s)] = cα,dE |X r+s|−(d−α) = cα,d (r + s)−(d−α)/βE |X 1|−(d−α)

E [Lt ,H(f )] = cα,d ,H∫ t

0

∫ t

0|r − s|2H−2(r + s)−(d−α)/βdrds

E [Lt ,H(f )] <∞ iff 2Hβ > d − αRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 15 / 21


Lemma (B. 2011)

If (Rα,H f )(0) <∞ for any α > 0, then Lt ,H(f ) <∞ for all t > 0 a.s.Moreover,

lim supt→∞

1t

log Lt ,H(f ) ≤ 0 a.s.

Example

(Xt )t≥0 is a symmetric β-stable process, β ∈ (0,2].f (x) = cα,d |x |−(d−α) with α ∈ (0,d).Since (Xt )t≥0 is self-similar with exponent 1/β,

X r+sd= (r + s)1/βX 1

E [f (X r+s)] = cα,dE |X r+s|−(d−α) = cα,d (r + s)−(d−α)/βE |X 1|−(d−α)

E [Lt ,H(f )] = cα,d ,H∫ t

0

∫ t

0|r − s|2H−2(r + s)−(d−α)/βdrds

E [Lt ,H(f )] <∞ iff 2Hβ > d − αRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 15 / 21

A hyperbolic equation


The equation

∂2u∂t2 (t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (8)

with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). Assume:

Ψ(·) is real-valued

Remark: We may not be able to identify the fundamental solution G.

Definition (Foondun, Khoshnevisan, E. Nualart, 2010)

A weak solution of (8) is the process {u(t , ϕ); t > 0, ϕ ∈ S(Rd )}:

u(t , ϕ) =1

(2π)d

∫ t

0

∫Rd

sin(√

Ψ(ξ)(t − s))√Ψ(ξ)

Fϕ(ξ) W (ds,dξ)




The equation

∂2u∂t2 (t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (8)

with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). Assume:

Ψ(·) is real-valued

Remark: We may not be able to identify the fundamental solution G.

Definition (Foondun, Khoshnevisan, E. Nualart, 2010)

A weak solution of (8) is the process {u(t , ϕ); t > 0, ϕ ∈ S(Rd )}:

u(t , ϕ) =1

(2π)d

∫ t

0

∫Rd

sin(√

Ψ(ξ)(t − s))√Ψ(ξ)

Fϕ(ξ) W (ds,dξ)



The Fourier transform of W

Let E be the set of linear combinations of functions

h(t , x) = 1[0,a](t)ψ(x), a > 0, ψ ∈ S(Rd )

Let Fh(t , ξ) =∫Rd e−iξ·xh(t , x)dx . Endow E with the inner product

〈h1,h2〉H = 〈Fh1,Fh2〉H

For any h ∈ E , defineW (h) := W (Fh)

The Hilbert space H is the completion of E with respect to 〈·, ·〉H.The map h 7→ W (h) is extended to H

W (h) :=

∫ ∞0

∫Rd

h(t , ξ)W (dt ,dξ) (stochastic integral)



Proposition

Let h : R+ × Rd → C be such that h(t , ·) = 0 if t > T , h(t , ·) ∈ L2(Rd )for any t ∈ [0,T ] and g := Fh ∈ H. Then h ∈ H and

‖h‖2H = αH(2π)2d∫Rd

∫ T

0

∫ T

0|r − s|2H−2h(r , ξ)h(s, ξ)drdsµ(dξ)

ht ,ϕ(s, ξ) =1

(2π)d 1[0,t](s)sin(

√Ψ(ξ)(t − s))√

Ψ(ξ)Fϕ(ξ)

The weak solution of (8) exists since ht ,ϕ ∈ H for any t > 0, ϕ ∈ S(Rd )(by Theorem 2, gt ,ϕ := Fht ,ϕ ∈ H). Moreover,

E |u(t , ϕ)|2 = ‖ht ,ϕ‖2H =

∫Rd

Nt (ξ)|Fϕ(ξ)|2µ(dξ), where

Nt (ξ) =αH

Ψ(ξ)

∫ t

0

∫ t

0|r − s|2H−2 sin(r

√ψ(ξ)) sin(s

√Ψ(ξ))drds



Proposition

Let h : R+ × Rd → C be such that h(t , ·) = 0 if t > T , h(t , ·) ∈ L2(Rd )for any t ∈ [0,T ] and g := Fh ∈ H. Then h ∈ H and

‖h‖2H = αH(2π)2d∫Rd

∫ T

0

∫ T

0|r − s|2H−2h(r , ξ)h(s, ξ)drdsµ(dξ)

ht ,ϕ(s, ξ) =1

(2π)d 1[0,t](s)sin(

√Ψ(ξ)(t − s))√

Ψ(ξ)Fϕ(ξ)

The weak solution of (8) exists since ht ,ϕ ∈ H for any t > 0, ϕ ∈ S(Rd )(by Theorem 2, gt ,ϕ := Fht ,ϕ ∈ H). Moreover,

E |u(t , ϕ)|2 = ‖ht ,ϕ‖2H =

∫Rd

Nt (ξ)|Fϕ(ξ)|2µ(dξ), where

Nt (ξ) =αH

Ψ(ξ)

∫ t

0

∫ t

0|r − s|2H−2 sin(r

√ψ(ξ)) sin(s

√Ψ(ξ))drds



Random Field Solution

Defintion

Let M = ∩t>0Mt where Mt is the completion of S(Rd ) with respect to

‖ϕ‖t = E |u(t , ϕ)|2 =

∫Rd

Nt (ξ) |Fϕ(ξ)|2µ(dξ)

Equation (8) has a random-field solution u(t , x) := u(t , δx ) if

δx ∈ M for all x ∈ Rd

Let Z = ∩t>0Zt where Zt is the completion of S(Rd ) with respect to

|‖ϕ‖|t =

∫Rd

(1

1/t + Ψ(ξ)

)H+1/2

|Fϕ(ξ)|2µ(dξ)

Remark: For any s, t > 0, Zt = Zs.



Random Field Solution

Defintion

Let M = ∩t>0Mt where Mt is the completion of S(Rd ) with respect to

‖ϕ‖t = E |u(t , ϕ)|2 =

∫Rd

Nt (ξ) |Fϕ(ξ)|2µ(dξ)

Equation (8) has a random-field solution u(t , x) := u(t , δx ) if

δx ∈ M for all x ∈ Rd

Let Z = ∩t>0Zt where Zt is the completion of S(Rd ) with respect to

|‖ϕ‖|t =

∫Rd

(1

1/t + Ψ(ξ)

)H+1/2

|Fϕ(ξ)|2µ(dξ)

Remark: For any s, t > 0, Zt = Zs.



Theorem 5 (B. 2011)


c(1)H t

(1

1/t2 + Ψ(ξ)

)H+1/2

≤ Nt (ξ) ≤ c(2)H t

(1

1/t2 + Ψ(ξ)

)H+1/2

CorollaryWe have Mt = Zt2 for any t > 0, and M = Z. Equation (8) has arandom-field solution iff

δx ∈ Z for all x ∈ Rd .

A necessary and sufficient condition for this is:∫Rd

(1

1 + Ψ(ξ)

)H+1/2

µ(dξ) <∞.



Theorem 5 (B. 2011)


c(1)H t

(1

1/t2 + Ψ(ξ)

)H+1/2

≤ Nt (ξ) ≤ c(2)H t

(1

1/t2 + Ψ(ξ)

)H+1/2

CorollaryWe have Mt = Zt2 for any t > 0, and M = Z. Equation (8) has arandom-field solution iff

δx ∈ Z for all x ∈ Rd .

A necessary and sufficient condition for this is:∫Rd

(1

1 + Ψ(ξ)

)H+1/2

µ(dξ) <∞.



Thank you!


Documents

Some linear SPDEs with fractional noiseaix1.uottawa.ca/~rbalan/Banff-slides.pdfOutline 1 Linear SPDEs with fractional noise 2 A parabolic equation 3 A hyperbolic equation Raluca Balan