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arXiv:1201.5635v1 [math.PR] 26 Jan 2012

White Noise Representation of Gaussian Random Fields

Zachary Gelbaum

Department of Mathematics

Oregon State University

Corvallis, Oregon 97331-4605, USA

Abstract

We obtain a representation theorem for Banach space valued Gaussian random variables as

integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the

existence of a white noise representation for a Gaussian random field indexed by a compact measure

space. As an application we show how existing theory for integration with respect to Gaussian

processes indexed by [0,1] can be extended to Gaussian fields indexed by compact measure spaces.

Keywords: white noise representation, Gaussian random field, stochastic integral

1. Introduction

Much of literature regarding the representation of Gaussian processes as integrals against white

noise has focused on processes indexed by R, in particular canonical representations (most recently

see [8] and references therein) and Volterra processes (e.g. [1, 3]). An example of the use of such

integral representations is the construction of a stochastic calculus for Gaussian processes admitting

a white noise representation with a Volterra kernel (e.g. [1, 11]).

In this paper we study white noise representations for Gaussian random variables in Banach

spaces, focusing in particular on Gaussian random fields indexed by a compact measure space. We

show that the existence of a representation as an integral against a white noise on a Hilbert space

H is equivalent to the existence of a version of the field whose sample paths lie almost surely in

H. For example as a consequence of our results a centered Gaussian process Yt indexed by [0,1]

admits a representation

?1

0

Yt

d=

h(t,z)dW(z)

for some h ∈ L2([0,1]×[0,1],dν×dν), ν a measure on [0,1] and W the white noise on L2([0,1],dν)

if and only if there is a version of Ytwhose sample paths belong almost surely to L2([0,1],dν).

The stochastic integral for Volterra processes developed in [11] depends on the existence of a

white noise integral representation for the integrator. If there exists an integral representation for

a given Gaussian field then the method in [11] can be extended to define a stochastic integral with

respect to this field. We describe this extension for Gaussian random fields indexed by a compact

measure space whose sample paths are almost surely square integrable.

Email address: gelbaumz@math.oregonstate.edu (Zachary Gelbaum)

Preprint submitted to Elsevier

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Section 2 contains preliminaries we will need from Malliavin Calculus and the theory of Gaussian

measures over Banach spaces. In section 3, Theorem 1 gives our abstract representation theorem

and Corollary 2 specializes to Gaussian random fields indexed by a compact measure space. Section

4 contains the extension of results in [11].

2. Preliminaries

2.1. Malliavin Calculus

We collect here only those parts of the theory that we will explicitly use, see [15].

Definition 1. Suppose we have a Hilbert space H. Then there exists a complete probability space

(Ω,F,P) and a map W : H → L2(Ω,P) satisfying the following:

1. W(h) is a centered Gaussian random variable with E[W(h)2] = ?h?H

2. E[W(h1)W(h2)] = ?h1,h2?H

This process is unique up to distribution and is called the Isonormal or White Noise Process on H.

The classical example is H = L2[0,1] and W(h) is the Wiener-Ito integral of h ∈ L2.

Let S denote the set of random variables of the form

F = f(W(h1),...,W(hn))

for some f ∈ C∞(Rn) such that f and all its derivatives have at most polynomial growth at infinity.

For F ∈ S we define the derivative as

n

?

1

DF =

∂jf(W(h1),...,W(hn))hj.

We denote by D the closure of S with respect to the norm induced by the inner product

?F,G?D= E[FG] + E[?DF,DG?H].

(D is usually denoted D1,2.)

We also define a directional derivative for h ∈ H as

DhF = ?DF,h?H.

D is then a closed operator from L2(Ω) to L2(Ω,H) and dom(D) = D. Further, D is dense

in L2(Ω). Thus we can speak of the adjoint of D as an operator from L2(Ω,H) to L2(Ω). This

operator is called the divergence operator and denoted by δ.

dom(δ) is the set of all u ∈ L2(Ω,H) such that there exists a constant c (depending on u) with

|E[?DF,u?H]| ≤ c?F?

for all F ∈ D. For u ∈ dom(δ) δ(u) is characterized by

E[Fδ(u)] = E[?DF,u?H]

for all F ∈ D.

For examples and descriptions of the domain of δ see [15], section 1.3.1.

When we want to specify the isonormal process defining the divergence we write δW. We will

also use the following notations interchangeably

δW(u),

?

udW.

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2.2. Gaussian Measures on Banach Spaces

Here we collect the necessary facts regarding Gaussian measures on Banach spaces and related

notions that we will use in what follows. For proofs and further details see e.g. [4, 5, 6, 13, 16]. All

Banach spaces are assumed real and separable throughout.

Definition 2. Let B be a Banach space. A probability measure µ on the borel sigma field B of B

is called Gaussian if for every l ∈ B∗the random variable l(x) : (B,B,µ) → R is Gaussian. The

mean of µ is defined as

?

B

µ is called centered if m(µ) = 0. The (topological) support of µ in B, denoted B0, is defined as the

smallest closed subspace of B with µ-measure equal to 1.

m(µ) =

xdµ(x).

The mean of a Guassian measure is always an element of B, and thus it suffices to consider only

centered Gaussian measures as we can then acquire any Gaussian measure via a simple translation

of a centered one. For the remainder of the paper all measures considered are centered.

Definition 3. The covariance of a Gaussian measure is the bilinear form Cµ: B∗×B∗→ R given

by

Cµ(k,l) = E[k(X)l(X)] =

?

B

k(x)l(x)dµ(x).

Any gaussian measure is completely determined by its covariance: if for two Gaussian measures

µ, ν on B we have Cµ= Cνon B∗× B∗then µ = ν.

If H is a Hilbert space then

Cµ(f,g) = E[?X,f??X,g?] =

?

B

?x,f??x,g?dµ(x)

defines a continuous, positive, symmetric bilinear form on H × H and thus determines a positive

symmetric operator Kµ on H. Kµ is of trace class and is injective if and only if µ(H) = 1.

Conversely, any positive trace class operator on H uniquely determines a Guassian measure on H

[6]. Whenever we consider a Gaussian measure µ over a Hilbert space H we can after restriction

to a closed subspace assume µ(H) = 1 and do so throughout.

We will denote by Hµthe Reproducing Kernel Hilbert Space (RKHS) associated to a Gaussian

measure µ on B . There are various equivalent constructions of the RKHS. We follow [16] and refer

the interested reader there for complete details.

For any fixed l ∈ B∗, Cµ(l,·) ∈ B (this is a non trivial result in the theory). Consider the linear

span of these functions,

S = span{Cµ(l,·) : l ∈ B∗}.

Define an inner product on S as follows: if φ(·) =?n

< φ,ψ >Hµ≡

1

1aiCµ(li,·) and ψ(·) =?m

aibjCµ(li,kj).

1bjCµ(kj,·) then

n

?

m

?

1

Hµis defined to be the closure of S under the associated norm ?·?Hµ. This norm is stronger than

? · ?B, Hµis a dense subset of B0and Hµhas the reproducing property with reproducing kernel

Cµ(l,k):

?φ(·),Cµ(l,·)?Hµ= φ(l)∀l ∈ B∗, φ ∈ Hµ.

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Remark 1. Often one begins with a collection of random variables indexed by some set, {Yt}t∈T.

For example suppose (T,ν) is a finite measure space. Then setting K(s,t) = E[YsYt] and supposing

that application of Fubini-Tonelli is justified we have for f,g ∈ L2(T)

?

TT

E[?Y,f??Y,g?] =

?

E[Ys,Yt]f(s)g(t)dνdν = ?K(s,t)(f),g?

where we denote?

symmetric and trace class then the above collection {Yt}t∈T determines a measure µ on L2(T) and

the above construction goes through with Cµ(f,g) = ?K(s,t)(f),g? and the end result is the same

with Hµa space of functions over T.

TK(s,t)f(s)dν(s) by K(s,t)(f). If one verifies that this last operator is positive

Define HXto be the closed linear span of {X(l)}l∈B∗ in L2(Ω,P) with inner product

?X(l),X(l′)?HX= Cµ(l,l′) (again for simplicity assume X is nondegenerate). From the reproducing

property we can define a mapping RX from Hµto HX given initially on S by

RX(

n

?

1

ckCµ(lk,·)) =

k

?

1

ckX(l)

and extending to an isometry. This isometry defines the isonormal process on Hµ.

In the case that H is a Hilbert space and µ a Gaussian measure on H with covariance operator

K it is known that Hµ=√K(H) with inner product ?√K(x),√K(y)?Hµ= ?x,y?H.

It was shown in [12] that given a Banach space B there exists a Hilbert space H such that B is

continuously embedded as a dense subset of H. Any Gaussian measure µ on B uniquely extends

to a Gaussian measure µH on H. The converse question of whether a given Gaussian measure on

H restricts to a Gaussian measure on B is far more delicate. There are some known conditions e.g.

[7]. The particular case when X is a metric space and B = C(X) has been the subject of extensive

research [14]. Let us note here however that either µ(B) = 0 or µ(B) = 1 (an extension of the

classical zero-one law, see [4]).

From now on we will not distinguish between a measure µ on B and its unique extension to H

when it is clear which space we are considering.

3. White Noise Representation

3.1. The General Case

The setting is the following: B is a Banach space densely embedded in some Hilbert space H

(possibly with B = H), where H is identified with its dual, H = H∗. (A Hilbert space equal to its

dual in this way is called a Pivot Space, see [2]).

The classical definition of canonical representation has no immediate analogue for fields not

indexed by R, but the notion of strong representation does.

Then WX(h) = RX(L∗(h)) defines an isonormal process on H and σ({WX(h)}h∈H) = σ(HX) =

σ({X(l)}l∈B∗) where the last inequality follows from the density of H in B∗.

We now state our representation theorem.

Let L : Hµ → H be unitary.

Theorem 1. Let B be a Banach space, µ a Gaussian measure on B, and Cµthe covariance of µ

on B∗× B∗. Then µ is the distribution of a random variable in B given as a white noise integral

of the form

?

X(l) =

h(l)dW.

(3.1)

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for some h : B∗→ H and a Hilbert space H, where h|H is a Hilbert-Schmidt operator on H.

Moreover, the representation is strong in the following sense: σ({WX(h)}h∈H) = σ({X(l)}l∈B∗).

Proof. B ⊂ H = H∗as above. Let WX be the isonormal process constructed above and Cµ(l,k)

the covariance of µ. Let L be a unitary map from Hµto H and define the function kL(l) : B∗→ H

by

kL(l) ≡ L(Cµ(l,·)).

Consider the Gaussian random variable determined by

Y (l) ≡

?

kL(l)dWX.

We have

Cov(Y (l1),Y (l2)) = ?kL(l1),kL(l2)?H= ?Cµ(l1,·),Cµ(l2,·)?Hµ= Cµ(l1,l2)

so that µ is the distribution of Y (l) and

X(l)

d=

?

kL(l)dWX.

It is clear that kLis linear and if Cµ(h1,h2) = ?K(h1),h2?H, h1,h2∈ H, then from above

k∗

LkL= K.

Because K is trace class this implies that kLis Hilbert-Schmidt on H.

From the preceding discussion we have σ({WX(h)}h∈H) = σ({X(l)}l∈B∗).

Remark 2. While the statement of the above theorem is more general than is needed for most

applications, this generality serves to emphasize that having a “factorable” covariance and thus an

integral representation are basic properties of all Banach space valued Gaussian random variables.

Remark 3. The kernel h(l) is unique up to unitary equivalence on H, that is if L′= UL for some

unitary U on H L as above, then

?

hL′(l)dW

d=

?

U (hL(l))dW

d=

?

hL(l)dW.

Remark 4. In the proof above,

?kL(l1),kL(l2)?H= Cµ(l1,l2)(3.2)

is essentially the “canonical factorization” of the covariance operator given in [17], although in a

slightly different form.

Remark 5. In the language of stochastic partial differential equations, what we have shown is that

every Gaussian random variable in a Hilbert space H is the solution to the operator equation

L(X) = W

for some closed unbounded operator L on H with inverse given by a Hilbert-Schmidt operator on

H.

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3.2. Gaussian Random Fields

The proof of Theorem 1 has the following corollary for Gaussian random fields:

Corollary 2. Let X be a compact Hausdorff space, ν a positive Radon measure and H = L2(X,dν).

If {Bx} is a collection of centered Gaussian random variables indexed by X, then {Bx} has a version

with sample paths belonging almost surely H if and only if

Bx

d=

?

h(x,·)dW

(3.3)

for some h : X → H such that the operator K(f) ≡?

case (3.2) takes the form

Xh(x,z)f(z)dν(z) is Hilbert-Schmidt. In this

E[BxBy] =

?

X

h(x,z)h(y,z)dν(z).

In other words, the field Bx determines a Gaussian measure on L2(X,dν) if and only if Bx

admits an integral representation (3.3).

3.3. Some Consequences and Examples

In principle, all properties of a field are determined by its integral kernel. Without making an

exhaustive justification of this statement we give some examples:

In Corollary 2 above, being the kernel of a Hilbert-Schmidt operator, h ∈ L2(X × X,dν × dν).

This means that we can approximate h by smooth kernels (supposing these are available). If we

assume h(x,·) is continuous as a map from X to H i.e.

lim

x→y?h(x,·) − h(y,·)?H= 0

for each y ∈ X and let hn∈ C∞(X), hn

that if

L2

→ h it follows that ?hn(x,·) − h(x,·)?H→ 0 pointwise so

?

Bn

x=

hn(x,·)dW

we have

E[Bn

xBn

y] → E[BxBy]

point-wise. This last condition is equivalent to

Bn

d

→ B

and we can approximate in distribution any field over X with a continuous (as above) kernel by

fields with smooth kernels.

The kernel of a field over Rddescribes its local structure [9]: The limit in distribution of

lim

rn→0

cn→0

X(t + cnx) − X(t)

rn

is

lim

rn→0

cn→0

?

h(t + cnx) − h(t)

rn

dW

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where h is the integral kernel of X, and this last limit is determined by the limit in H of

lim

rn→0

cn→0

h(t + cnx) − h(t)

rn

.

The representation theorem yields a simple proof of the known series expansion using the RKHS.

The setting is the same as in Theorem 1.

Proposition 3. Let Y (l) be a centered Gaussian random variable in a Hilbert space H with integral

kernel h(l). Let {ek}∞

{ξk} such that

Y (l) =

?

1

1 be a basis for H. Then there exist i.i.d. standard normal random variables

∞

ξkΦk(l)

where Φk(l) = ?h(l),ek?H and the series converges in L2(Ω) and a.s.

Proof. For each l

h(l)=

∞

?

1

Φk(l)ek.

We have

Y (l) =

?

∞

?

1

Φk(l)ekdW =

∞

?

1

Φk(l)ξk

where {ξk} = {?ekdW} are i.i.d. standard normal as

{Φk(l)} ∈ l2(N) the series converges a.s. by the martingale convergence theorem.

?dW is unitary from H to L2(Ω). As

4. Stochastic Integration

Combined with Theorem 1 above, [11] furnishes a theory of stochastic integration for Gaussian

processes and fields, which we now describe for the case of a random field with square integrable

sample paths as in Corollary 2.

Denote by µ the distribution of {Bx} in H = L2(X,dν) and as above the RKHS of Bx by

Hµ⊂ H. Let

Bx=

h(x,·)dW

?

and L∗(f) =?h(x,y)f(y)dν(y). Then L∗: H → Hµis an isometry and the map v ?→ RB(L∗(v)) ≡

W(v) : H → HB (HB is the closed linear span of {Bx} as defined in sec. 2) defines an isonormal

process on H. Denote this particular process by W in what follows.

First note that as Hµ= L∗(H) and L is unitary, it follows immediately that D1,2

where we use the notation in [15, 11] and the subscript indicates the underlying Hilbert space.

Hµ= L∗(D1,2

H)

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The following proof from [11] carriesover directly: For a smooth variable F(h) = f(B(L∗(h1),...,B(L∗(hn))

we have

E?DB(F),u?Hµ=E?

n

?

1

f′(B(L∗(h1),...,B(L∗(hn))L∗(hk),u?Hµ

= E?

= E?

= E?DW(F),L(u)?H

?

?

f′(B(L∗(h1),...,B(L∗(hn))hk,L(u)?H

f′(W(h1),...,W(hn))hk,L(u)?H

which establishes

dom(δB) = L∗(dom(δW))

and

?

udB =

?

L(u)dW.

The series approximation in [11] also extends directly to this setting.

Theorem 4. If {Φk} is a basis of Hµthen there exists i.i.d. standard normal {ξk} such that:

1. If f ∈ Hµand

?

fdB =

∞

?

1

?f,Φk?Hµξk

a.s.

2. If u ∈ DHµthen

?

udB =

∞

?

1

(?u,Φk?Hµ− ?DB

Φku,Φk?Hµ)

a.s.

Proof. The proof of (1) and (2) follows that in [11].

Remark 6. For our purposes the method of approximation via series expansions above seems

most appropriate. However in [1] a Riemann sum approximation is given under certain regularity

hypotheses on the integral kernel of the process, and this could be extended in various situations

as well.

Remark 7. The availability of the kernel above suggests the method in [1] whereby conditions are

imposed on the kernel in order to prove an Ito Formula as promising for extension to more general

settings.

5. References

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