Random fields and the geometry of Wiener space\thanksrefT1

Random fields and the geometry of Wiener space\thanksrefT1

[ [    [ [ Stanford University and TIFR-CAM Department of Statistics
Sequoia Hall
Stanford University
390 Serra Mall
Stanford, California 94305-4065
USA
\printeade1
TIFR—Center for Applicable Mathematics
Sharadanagar, Chikkabommasandra
Post Bag 6503, GKVK Post Office
Bangalore 560 065
India
\printeade2
\smonth5 \syear2011\smonth10 \syear2011
\smonth5 \syear2011\smonth10 \syear2011
\smonth5 \syear2011\smonth10 \syear2011
Abstract

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set under the standard Gaussian law . Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.

[
\kwd
\doi

10.1214/11-AOP730 \volume41 \issue4 2013 \firstpage2724 \lastpage2754 \setattributeabstract width 287pt \newproclaimdefinition[theorem]Definition \newproclaimremark[theorem]Remark \newproclaimExample[theorem]Example

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Random fields and the Wiener space \thankstextT1Supported in part by NSF Grants DMS-09-06801 and DMS-04-05970.

{aug}

A]\fnmsJonathan E. \snmTaylorlabel=e1]jonathan.taylor@stanford.edu and B]\fnmsSreekar \snmVadlamani\correflabel=e2]sreekar@math.tifrbng.res.in

class=AMS] \kwd[Primary ]60G60 \kwd60H05 \kwd60H07 \kwd[; secondary ]53C65.

Wiener space \kwdMalliavin calculus \kwdrandom fields.

1 Introduction and motivation

We start with a description of a certain class of set functionals determined by the canonical Gaussian measure on . By canonical, we shall mean centered and having covariance . Its density with respect to the Lebesgue measure on is therefore given by . For this measure, we consider computing the probability content of a tube around , leading us to a Gaussian tube formula which we state as

(1)

where is the th Gaussian Minkowski Functional (GMF) of the set . If  is compact and convex, that is, if is a convex body, then we can take the right-hand side (1) to be a power series expansion for the left-hand side. For certain , this expansion must be taken to be a formal expansion, in the sense that up to terms of some order, the left and right-hand side above agree. For example, if  is a centrally-symmetric cone such as the rejection region for a or statistic, then has a singularity at the origin in the sense that the geometric structure of the cone around 0 is nonconvex and the expansion above is accurate only up to terms of size .

Our interest in this tube formula lies in the appearance of these coefficients in the expected Euler characteristic heuristic RFG (), Worsley1 (), Worsley3 (), Worsley2 ().

1.1 Expected Euler characteristic heuristic

The Euler characteristic heuristic was developed by Robert Adler and Keith Worsley (cf., e.g., RFG (), Worsley1 (), Worsley3 (), Worsley2 ()) to approximate the probability

with , where , and is the Euler–Poincaré characteristic.

Let be an -dimensional reasonably smooth manifold, with identically and independently distributed copies of a Gaussian random field defined on . Subsequently, for any , with two continuous derivatives, we can define a new random field on given by , for each .

Using the above Euler characteristic heuristic for approximating the -value for appropriately large values of , and Theorem of RFG (), we have

where for are the GMFs of the set that appear in (1), and for are the Lipschitz–Killing curvatures (LKCs) of the manifold defined with respect to the Riemannian metric given by , where and are two vector fields on , with and representing the directional derivatives of .

1.2 Curvature measures

The LKCs for a large class of subsets of any finite dimensional Euclidean space can be defined via a Euclidean tube formula. In particular, let be an -dimensional set with convex support cone, then writing as the standard -dimensional Euclidean measure, as the -dimensional unit ball centered at origin, for small enough values of , we have

where is the th LKC of the set with respect to the usual Euclidean metric, and ’s are called the Minkowski functionals of the set .

Geometrically, for a smooth -dimensional manifold embedded in is the -dimensional Lebesgue measure of the set , and the other LKCs can be defined as

where is the surface area of a unit ball in , are the principal curvatures at , and is the th symmetric polynomial in indices. In the case when the set is not unit codimensional, then the definition involves another integral over the normal bundle.

The (generalized) curvature measures defined this way are therefore signed measures induced by the Lebesgue measure of the ambient space. By replacing the Lebesgue measure in (1.2) with an appropriate Gaussian measure, we can define a parallel Gaussian theory. The GMFs in (1) play the role of Minkowski functionals in the Gaussian theory. In particular,

(4)

where is the integral of with respect to the th generalized Minkowski curvature measure, and is the th Hermite polynomial in (cf. RFG ()).

1.3 Our object of study: A richer class of random fields

In this paper we intend to extend (1.1) to a larger class of random fields , which can be expressed using , where is the space of continuous functions , such that , also referred to as the classical Wiener space, when equipped with the standard Wiener measure on this sample space. In other words, we shall consider random fields which can be expressed as some smooth Wiener functional. For instance, let us start with a smooth manifold together with a Gaussian field defined on it, such that its covariance function is given by

(5)

where is assumed to be a smooth function, with more details appearing in Section 6, where we actually prove an extension of (1.1). This infinite dimensional random field can be used to construct many more random fields on , for instance, the following: {Example}[(Stochastic integrals)] Let be a smooth function, and consider the following random field,

(6)

where is the Wiener functional

This is clearly an extension of the random fields in (1.1). As a consequence of our extension of the Gaussian Minkowski functionals to smooth Wiener functionals, we prove that, under suitable smoothness conditions on ,

Our smoothness conditions are rather strong in this paper: we assume is with essentially polynomial growth. We need such strict assumptions to ensure regularity of various conditional densities derived from the random field (6) and its first two derivatives at a point .

A quick look at (1.1) reveals that in order to extend it to the case when , we must be able to define GMFs for infinite dimensional subsets of , as . In the present form, that is, (4), the definition of GMFs appears to depend on the summability of the principal curvatures of the set at each point as well as the integrability of these sums. In infinite dimensions this summability requirement is equivalent to an operator being trace class. This is quite a strong requirement, and may be very hard to check. Indeed, the natural summability requirements of operators in the natural infinite dimensional calculus on , the Malliavin calculus, is the Hilbert–Schmidt class rather than the trace class.

Therefore, we shall first modify the definition of GMFs, from (4) to one which is more amenable for an extension to the infinite dimensional case. This will be done in Section 2.

After setting up the notation and some technical background on the Wiener space in Section 3, the all important step, that of extending the appropriate definition of GMFs to the case of codimension one, smooth subsets of the Wiener space, is accomplished in Section 5. The characterization of GMFs in the infinite dimensional case will be done precisely the same way as in the case of finite dimensions, where, as noted earlier, the GMFs are identified as the coefficients appearing in the Gaussian tube formula.

Finally, in Section 6, we use the infinite dimensional extension of the GMFs to obtain an extension of (1.1), for random fields which can be expressed as stochastic integrals driven by as defined in Example 1.3, and discuss other possible implications of the extension. Most of our methods in Section 6 are invariant to the formulation of the random field as a stochastic integral. Hence, should a random field satisfy all the regularity conditions appearing in Section 6, we expect our methods to work, modulo a few changes.

2 Preliminaries I: The finite dimensional theory

In this section we shall use the standard finite dimensional theory of transformation of measure for Gaussian spaces to modify the definition (4) of the GMFs to one which is more suited to extension to the infinite dimensional case.

We begin by recalling some well-known facts about analysis on finite dimensional Gaussian spaces from Section of Chapter II of Malliavin (), and Chapter  of UstZakai00 (). Let be the Gaussian measure on given by, and a mapping from into itself, given by , where is Sobolev differentiable and for any with . Then, the Radon–Nikodym derivative of with respect to the measure is given by

(7)

where is the usual Euclidean norm, and is the generalized Carleman–Fredholm determinant.

Subsequently, for a smooth, unit codimensional, convex set , let us define the tube of width around the set as the set , where is the -dimensional ball of radius centered at the origin. Next, we shall define a signed distance function given by

where denotes the interior of the set .

Applying the co-area formula, and using the fact that , we get

(8)

For fixed, we can now use equation (7) with any suitable transformation that agrees with on for some small positive . Any such transformation maps to for and any . Two further applications of the co-area formula yield

Therefore, equation (8) simplifies to

(9)

Using a yet-to-be justified Taylor series expansion of the integrand appearing in the above integral, we can finally rewrite the GMFs as

(10)

where is the outward unit normal vector field to the set , and is the surface measure of the set . Note that in the above expression we have removed the modulus around the part, which can be justified by taking reasonably small values of . This new definition of GMFs involves terms which have obvious extensions in the infinite dimensional case.

3 Preliminaries II: The infinite dimensional theory

In this section we recall some established concepts in Malliavin calculus which we shall need in later sections. We begin with an abstract Wiener space , where , equipped with the inner product , is a separable Hilbert space, called the Cameron–Martin space, is a Banach space into which is injected continuously and densely, and, finally, is the standard cylindrical Gaussian measure on . For the sake of simplicity, one can appeal to the classical case when we have as the space of real-valued, absolutely continuous functions on with derivatives, which is continuously embedded in the space of real-valued continuous functions on , such that .

Sobolev spaces on Wiener space

Following the notation used in Malliavin (), Nualart-book (), Watanabe93 (), Sobolev spaces for and are defined as the class of -valued functions such that

where is the Ornstein–Uhlenbeck operator defined on the Wiener space. Writing  as the Gross–Sobolev derivative and as its dual under the Wiener measure, . The Sobolev spaces for are the spaces of distributions, defined as the dual of , where, as usual, . Throughout this paper, whenever appropriate, we will adopt this convention.

The space of infinitely integrable, -smooth Wiener functionals is given by

Consequently, let us define the analogous infinitely integrable random variables as Finally, we shall end this section with another definition which translates to the regularity of Wiener functionals. {definition} For an -valued Wiener functional , the Malliavin covariance (matrix) , and the functional itself, is called nondegenerate in the sense of Malliavin if , whenever is well defined.

H-Convexity

In order to characterize the class of subsets of the Wiener space for which we shall define the GMFs, we shall recall the notion of -convexity.

{definition}

An -convex functional is defined as a measurable functional such that for any ,

(11)

One of the properties of -convex functionals which will be used in later sections is that a necessary and sufficient condition for a Wiener functional for some to be -convex is that the corresponding must be a positive and symmetric Hilbert–Schmidt operator valued distribution on (cf. UstZakai00 ()).

3.1 Quasi-sure analysis

In this section, most of which is based upon Malliavin (), Takeda (), we shall resolve some technical aspects of defining integrals of Wiener functionals with respect to measures concentrated on -zero sets. Since all Wiener functionals are de facto defined up to -zero sets, thus, in order to be able to define the integral of Wiener functionals with respect to measures which are concentrated on -zero sets, we must resort to what is referred to as quasi-sure analysis, which in turn relies on the concept of capacities on the Wiener space.

{definition}

Let and . For an open set of , we define its -capacity by

For each subset of of , we define its -capacity by

These capacities are finer scales to estimate the size of sets in than . In particular, a set of -capacity zero is always a -zero set, but the converse is not true in general.

A property is said to be true -quasi-everywhere (q.e.) if

One of the most crucial steps in obtaining the co-area formula in the Wiener space, which in turn is a necessary step to obtain the tube-formula in the Wiener space, is to be able to extend ordinary Wiener functionals to sets of -zero measure. Quasi-sure analysis lets us do precisely that and much more.

{definition}

A measurable functional is said to have a -redefinition , satisfying -almost surely, and is -quasi-continuous, if for all , there exists an open set of , such that and the restriction of to the complement set is continuous under the norm of uniform convergence on .

It can easily be seen that two redefinitions of the same functional differ only on a set of -capacity zero, thereby implying the uniqueness of a -redefinition up to -capacity zero sets. According to Theorem  of Malliavin (), every functional has a -quasi-continuous redefinition, which can be taken to be in the first Baire class.

In what follows in the remainder of this section, we recall some facts from the Malliavin calculus that will be helpful in our description of a tube below. If , one can make a statement similar to Theorem of Malliavin () related to the differentiability of , essentially a form of Taylor’s theorem with remainder.

Lemma 3.1

Suppose . Then, for each

for any .

{pf}

Define

where is the inverse of the Cauchy operator Malliavin (). For each , converges in , so the Kree–Meyer inequalities imply that also converges in . A second application of the Kree–Meyer inequalities implies that

Or, is Cauchy in , hence, its limit .

Hence, by the Borel–Cantelli property for the capacities (Corollary IV.1.2.4 of Malliavin ()), for each we can extract a sequence such that

(12)
Corollary 3.2

Suppose is nondegenerate and is a countable dense subset. Then,

(13)
{pf}

The only thing that needs verifying beyond what was pointed out above is that This follows from the Tchebycheff inequality (Theorem IV.2.2 of Malliavin ()) applied to and a Borel–Cantelli argument.

There is an obvious higher order Taylor expansion of , which we will use upto the second order term in our description of the tube below. If we are willing to sacrifice some moments, we can further specify in Corollary 3.2 that the existence of the partial derivatives of as a limit at implies their existence as limits at for all for some fixed, large .

Corollary 3.3

Suppose and is a countable dense subset. Then, for all

(14)
{pf}

This follows from the fact that the translation operator is a continuous map from to for any which follows directly from the Cameron–Martin theorem.

{remark}

Finally, we note that we can, by choosing appropriately, choose the set, say, in Corollary 3.3, in such a way that and for all and for all in some countable dense subset of .

4 Key ingredients for a tube formula

In this section we shall adopt a step-wise approach to reach our first goal, that of obtaining a (Gaussian) volume of the tube formula, for reasonably smooth subsets of the Wiener space. The three main steps are as follows: (i) characterizing subsets of the Wiener space via Wiener functionals, for which tubes, and thus GMFs, are well defined; (ii) assurance that the surface measures are well defined for the sets defined via the Wiener functionals; and finally, (iii) a change of measure formula for surface area measures corresponding to the lower dimensional surfaces of the Wiener space.

We shall first characterize the functionals for which the surfaces measures are well defined, subsequently, we shall prove a change of measure formula for the surfaces defined via such functionals. Finally, we shall define the class of sets for which the tube formula and GMFs are well defined by imposing more regularity conditions on the Wiener functionals.

4.1 The Wiener surface measures

Let us start with a reasonably smooth, valued Wiener functional . For , we write . The sets define a foliation of hypersurfaces imbedded in .

The surface measures of these foliations are closely related to the density of the push-forward measure on with respect to the Lebesgue measure on .

Heuristically, writing for the Dirac delta at , the density can be defined as

(15)

as long as we can make sense of the composition . For a smooth, real-valued Wiener functional , we also expect the following relation to hold:

(16)

where is the conditional expectation of given , assuming the composition is well defined.

Making this heuristic calculation rigorous leads us back to the Sobolev spaces of Section 3, where the object is related to a generalized Wiener functional, that is, an element of some for through the pairing

representing conditional expectation given for any . What is left to determine is, for a given , which Sobolev spaces contain .

The following theorem, the proof of which can be found in Watanabe93 (), provides the answer, taking us one step closer to defining the surface measure corresponding to the conditional expectation.

Theorem 4.1

Let be an -valued, nondegenerate Wiener functional such that for , and the density of the law of is bounded. Also, let and satisfy

(17)

and, finally, . Then for , with , we have

(18)

where is the Sobolev space of real-valued, weak -differentiable functions which are -integrable.

Recall that for , the density (cf. Nualart-book ()). Now using the differentiability of the density together with equations (16), (18) and the algebraic structure of the Sobolev spaces, we have for any , where .

That is, for each , there exists a continuous mapping This, in turn, induces a dual map defined via the dual relationship

(19)

Informally, this map, sometimes referred to as the Watanabe map (see Section 6 of Chapter III of Malliavin ()), is just composition, that is,

The object is almost the surface measure needed in (10), but it is just a generalized Wiener functional, that is, distribution on , at this point. If we are to justify our Taylor series expansion via a dominated convergence argument, we need to know that it has a representation as a measure on .

Clearly, for positive , we shall have

Therefore, defines a positive generalized Wiener functional. Next, Theorem 4.3 of Sugita () together with the conditions stated in Theorem 4.1 implies that for each , there exists a finite positive Borel measure defined on Borel subsets of the Wiener space , supported on , such that

for all , with its -quasi continuous redefinition.

The measure defined is a probability measure on the set . Using Airault and Malliavin’s arguments in AiraultMalliavin (), an appropriate area measure , corresponding to the measure , can be defined as

(20)

where is the Malliavin covariance matrix. Note that the surface measure depends only on the geometry of the set , whereas the conditional probability measure depends on the functional from which the set is derived, thus the superscripts on the respective measures. We are now in a position to justify at least part of (10).

Theorem 4.2

Let be a -valued nondegenerate Wiener functional such that and the density of the law of is bounded. Define the unit normal vector field . Furthermore, suppose that:

  • for in some neighborhood of 0;

  • for in some neighborhood of 0.

Then, for for some nonzero critical radius

(21)

Before proving the above theorem, we shall state a few results concerning the regularity of functions of smooth Wiener functionals.

Proposition 4.3

Let and .

  • If , then where for .

  • If almost surely and , then where for .

We shall skip the proofs of the above, as these can be proved by replicating the proofs of Theorems and of Watanabe Watanabe93 ().

{pf*}

Proof of Theorem 4.2 This is just dominated convergence combined with the nondegeneracy of as well as the following bound (cf. Theorem 9.2 of Simon ()):

for some fixed .

Note that while using the dominated convergence, we are inherently assuming the well definedness of integrals of and with respect to the surface measure , which requires

(22)
(23)

Now, using Theorem of Watanabe93 (), we have for all . Subsequently, using the above proposition together with the assumption involving the existence of exponential moments, we have such that where . In order to satisfy (22) and (23), we must choose and such that .

{remark}

Note that Theorem 4.2 does not say that the Gaussian measure of the tube is given by the power series in (4.2). Rather, it gives conditions on the sets for which the coefficients in the power series are well defined. These conditions allow us to define GMFs for level sets of functions that are not necessarily -convex. However, for such functions we will lose the interpretation of the power series in (4.2) as an expansion for the Gaussian measure of the tube. This is similar to the distinction between the formal and exact versions of the Weyl/Steiner tube formulae TakemuraKurikiEquivalence2002 ().

4.2 Change of measure formula: A Ramer type formula for surface measures

After assuring ourselves of the existence of the surface Wiener measures, we shall now move onto proving a change of measure formula for the surface measures given by equation (20).

To begin with, let so that we can define the surface measure using Theorem 4.1. In order to obtain a change of measure formula for the lower-dimensional subspaces of the Wiener space, we shall start with the standard change of measure formula on the Wiener space . Let us define a mapping given by , for some smooth . Moreover, let be an open subset of , and: {longlist}[(1)]

is a homeomorphism of onto an open subset of ,

is an -valued map and its derivative at each is a Hilbert–Schimdt operator on . This transformation induces two types of changes on the initial measure defined on . These two induced measures can be expressed as

for a Borel set of .

Ramer’s formula for change of measure on , induced by a transformation defined on and satisfying the above conditions, gives an expression for the Radon–Nikodym derivative of with respect to and can be stated as follows:

(24)

where denotes the Malliavin divergence of an -valued vector field in . The proof of this result can be found in Ramer (), UstZakai00 (). It is to be noted here that, for appropriately smooth transformations, a similar result for can be obtained by using the relationship between and given by

The following theorem is the first step toward obtaining similar formulae for change of measure on lower-dimensional subsets of the Wiener space.

Theorem 4.4

Let satisfy the conditions from Theorem 4.1, and and be as given in (17). Then, there exists a sequence of probability measures defined on Borel subsets of such that the measures are absolutely continuous with respect to the Wiener measure and the sequence converges weakly to .

{pf}

Let us choose a sequence of positive distributions on given by , such that it converges to weakly in and that for all . Then define the measures as

for all measurable on . In view of (19), we can clearly identify the restriction of the measures to with . Now, by construction, converges to in and their limit is a nonnegative generalized Wiener functional. Therefore, using Lemma 4.1 of Sugita (), we see that the measures converge weakly to .

Thus, the surface (probability) measure of , or the conditional probability measure corresponding to , for any can also be defined as

for the appropriate class of Wiener functionals , which, as noted earlier, depends on the regularity of .

Let us now define a mapping