Transformation of gaussian measures

© Annales mathématiques Blaise Pascal, 1996, tous droits réservés. L’accès aux archives de la revue « Annales mathématiques Blaise Pascal » (http: //math.univ-bpclermont.fr/ambp/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.


Introduction
We shall be, in our lecture, mainly concerned by some particular cases of the following problem : Let (X, F, ) be a measure space and T : X ~ X measurable. We denote by T( ) Example 2 : Let (SZ,.~',P) be the classical Wiener space, Q = Go((0,1~),.~ the Borel 03C3-field, P the Wiener measure. Let u : : [0, Ij x S2 -j IR, be a measurable and adapted stochastic process such that / oo almost surely, and let T : 03A9 ~ Q be defined by : = wt + / ds.
That is Q o T-1 ==P. . (This fact was first proven by means of the Itô-calculus, but as we shall see, we can obtain this by analytic methods).
This has an application in Statistical Communication Theory : Suppose we are receiving a signal corrupted by noise, and we wish to determine if there is indeed a signal or if we are just receiving noise.
If is the received signal, the noise and s(t) the emitted signal : : )=~)+~) (A) In general, we make an hypothesis on the noise : it is a white noise.
The "integrated~ version of (A) is X(t) = t 0 s ( u ) du + Wt = St + Wt (A') (W is the standard Wiener process, X(t) ck is the cumulative received signal). Now we ask the question : is there a signal corrupted by noise, or is there just a noise (~)=0, Vt)?
We consider the likelihood ratio and we fix a threshold level for the type I-error : if : dw dx (03C9) A we reject (H0) if : dw dx (03C9) ~ 03BB we accept (H0).

Some general considerations and examples.
If P « Q, then T(P) G T(Q).
(a) Therefore, we do not lose very much if we suppose that P and Q are probabilities.
In the case where Q is a probability, we can have an expression o f d T ( P ) d T ( Q ) as conditional mathematical expectation.
Remark : From (a) we see that, if there exists a probability Q such that P « Q and T(Q) = P, then T(P) G P.
The converse is true if moreover > 0. (The measures are equivalent). Therefore the following properties are equivalent : (z) : T(P) ^' P, (ii) : : 3Q -P such that T(Q) = P.
Let us now consider an example which allows us to guess the situation in infinite dimensional space.

The problem is :
"what does the word perturbation mean ?" CHAPTER ONE Anticipative stochastic integral 1 -Gaussian measures on Banach spaces Let E be a (real) separable Banach space, E' its dual. A (Borelian) probability õ n E is said to be "Gaussian centered" if for every ~' E E', (., ~~)E,E~ = z'(.) is a Gaussian centered (real) variable (eventually degenerated) under ~c. All what we shall say is true whatever be the dimension of E (finite or infinite).
(Bochner integral of a vector function). It is the barycenter of the measure (., x'~d . A is injective if Supp p = E . Let a; E A(E') so x = A(u') and let y E A(E') so y = A(v') , we shall put on A(E') C E the following scalar product : (it does not depend on u' and v').
Actually : Let H~ be the completion of A(E') with respect to ~~.~~~,. We have i : H~, --> E. I say that z is injective (it will allow us to consider H~, as a subspace of E). H~, is called the "reprodncang kernel Hilbert space" (r.k.H.s.) of u.
By the generalized integration by parts this is equal to : : 1) H is separable, as a Hilbert space. Therefore it is a borelian subset of E , 2) p,(H) = 0 or I and p,(H) = 0 ~ dimH = +~ (therefore p,(H) =1 b dimH oo), 3) H is the intersection of the family of measurable subspaces of E, whose probability is equal to one, 4) the canonical injection i : H -~ E is compact, 5) for every Hilbert space K and u : : .E 2014~ K linear continuous, U o z : : H K is Hilbert-Schmidt, 6) for every Hilbert space K and v :.K --~ E' linear continuous, i* o v : : K --~ H is Hilbert-Schmidt.
2 -L2-functionals on an abstract Wiener space Let (H, E, /~) be an abstract Wiener space. Suppose (ej is a sequence of elements of E' such that (i* is an orthonormal basis in H. A function f : E ---~ IR, is said to be a polynomial in the (ej) if there exists an integer n and a polynomial function P on IRn such that f ( x) = (x), ..., en(x)), ~x E E.
We denote deg f := deg P (P is not defined uniquely but the degree of f is independent of the choice of P).
We denote by P((ej )) the set of polynomials and by the set of polynomials of degree n. It is easy to see that is contained in each ,Cp(E, 0 p oo (but clearly not in Moreover, is dense in for these p. Therefore, independent of the chosen orthonormal family (e~). The same is true for each°~( (e~~)~ . Example : If n = 1, is the family of afline continuous functions : an element of is a linear continuous function on E plus a constant.
We have : = H C IR (see infra).
We call pn L2 the set of generalized polynomials . of degree' at most n is a Hilbert space.
Let us now introduce the "Wiener chaos decomposition" (or "Wiener-Itô decomposition"). Let Co = P0L2 the vector space of ( -equivalence classes of ) constants.
We define Cn inductively as follows : Cn is the orthogonal complement in ~'~ L2 of (Co ~ ... ® Cn _ 1 ) . (In other words Cn is the set of generalized polynomials of degree n, orthogonal to all generalized polynomials of degree less than n). The Cn are called the "nth chaos" ( or "chaos of order n"). Cl is isomorphic to H. We have a description of elements of Cn in term of Hermite polynomials.
We recall that the Hermite polynomials in one variable are defined by : e x p { t 2 2 } d n d t n ( e x p { -t 2 2 } ) , A linear mapping f : : E ~ IR is said to be a "linear measurable functional" if there exists a sequence of linear continuous functionals on E, converging to f , p-almost surely.
If x E H, it defines a linear measurable functional x(.). Actually, if xn is a sequence of elements of E' C H such that zn --~ x in H, then xn(.) converges to the random variablẽ defined by x, in L2(E, ~c). Therefore,there exists a subsequence converging almost surely to x. Moreover, oo . ' The converse is true, shown by the following proposition .
If h E H, the random variable h on E will be denoted by x ~ (x, h)H. Proposition : : Every linear measurable functional, 1, has a restriction to H which is continuous (for the Hilbertian topology). If we denote by f o this restriction we have .
The converse is true.

Proof :
We have already noticed that the converse is true. Let Take £ the linear subspace generated by A, we see that the above convergence holds for all x Since ~c(~) = 1, then H and therefore Therefore the restriction of f to H is uniquely defined. Therefore (xn (.)) converges in L2 (~c Moreover, we have : : Let (03C6j) be an orthonormal basis of K.
We have :~A If we integrate term by term these equalities, we obtain : Conversely let B E GZ(H, K). We have for x E H : : .7 Now each term in the right-hand member possesses a linear measurable continuation to E, and the series converges in LZ (E, ~c. K).
We have then defined a linear measurable extension of A to E. -Q.E.D.

-Derivatives of functionals on a Wiener space
Let (E, H, be an abstract Wiener space and let K be another Hilbert space. Let f : E -~ K be a function. ' We say that f possesses a Fréchet derivative in the direction of H, at the point (Analogous assertion for generalized polynomials, or "moderate" regular functions Pj ) . The formula needs an explanation : In the right member is the indefinite integral null at zero of I~ _ 1 ( f n ) (w ) : J ( I n -1 ( f t n ) ) = t 0 In-1(fsn) ds .
Therefore VF(w) is an element of the Cameron-Martin space.
We now give several useful properties : . The set of polynomial functions on E, with values in K is dense in ID2,1(K).
. Therefore the algebraic sum of chaos ~ Cn is dense in ~ 2 ~ 1, . The set of smooth functions on E is dense in C2~~ (a function is said to be "smooth " if it has the form : with f belonging to f and its derivatives are bounded).
. Let 03C6 : IRn ~ IR be a function in C1b(IRn) and let F1, ..., Fn E ID2,1. Then This result is false if the above hypothesis is not satisfied. For instance on IR, f ---9 = eX E ID2,1, but f o g ~ L 2 ( I R n , 0 3 B 3 n ) .
Remark : The operator V, called the "stochastic" gradient, or "stochastic" derivative, is very close to the ordinary gradient as we can see. The usual gradient at the point Xo is an element of E' (if the function takes its values in IR). The stochastic gradient is the composite of the ordinary gradient by the application i* from E' to H.
In an analogous manner if f : : E -K has an ordinary gradient, this gradient is a linear mapping of E into K ; f ' E --~ K.
The transpose t f' is a linear continuous mapping from K into E' .

-Anticipative stochastic integral
Definition : : TAe transpose of the opener V M called the "Skorokhod integral", or the "divergence operator".  In particular, if G : E ~ H is such that G(x) = a, ~x : sG = ~(:).

If F E L2 (E. H) is such that, for every a E H : :
(F, a)H 5(.) belongs to L2(E, , we shall say that F is "2-Ogawa integrable" when there exists G E L2 (E, such that (F, Pn) H ---~ G in quadratic mean.
(The Pn being as above).
In this case , we have : For the proof see Nualart-Pardoux. This results in a localization theorem : if F is null (almost everywhere) on a set, so is its derivative. The derivation is a "local operator". ( Note that 6(F) may depend on the localizing sequence .
We shall need another notion of stochastic derivatives and Skorokhod integrals for some functions not necessarily belonging to ID2,1, nor Skorokhod integrable, introduced by Buckdahn : Let T : E --~ E be a measurable mapping of the form : Let ~ E and suppose that for every sequence of smooth random variables {~'~) E ID2,1 converging to £ in ~2'~, the following limit exists and is independent of the approximating sequence chosen : lim V (~n a ~') n 2014~ oo where the limit is taken in probability.
Let us remark that 03BEn o T belongs to ID2,1 since the 03BEn are smooth.
The common limit of the above sequences is denoted by V (~ o T).  if the limit of the right member is taken in probability.
( ~ei denotes the generalized derivative in the ei-direction introduced just above).
Lemma 3 : : Suppose T = I + F as above is such that « ~c. Then b(~' o T) exists and satisfies the following identity : : 1 -Preliminary results on equivalence and orthogonality of product measures Let (Ek, Bk)k~IN* be a sequence of measurable spaces and for every k, let k and vk be two probabilities on (Ek, Bk) such that k « Vk . Let us set Pk = d k. If this condition is not satisfied we have orthogonality.

-Affine transformations of Gaussian measures
Now let be an abstract Wiener space. If (en) is an orthonormal basis of H, the random variables en are independent Gaussian variables on E, with mean zero and variance one. The law of the sequence (en) is therefore a product measure on IRIN : 00 03B3IN = ~ 03B3n n=O where Tn = T (Gaussian measure on IR) for every n. Now we have a measurable (defined almost everywhere) map 03B8 of E into IRIN : x ^^~ (en(x))~.
If the en belong to E', the en are everywhere defined and 0 is continuous from E into IRIN. It is clear now that the image of under 0 is equal to 03B3IN . We have 0(H) = l2 as we can see immediately (the en{x) are defined in a unique way on H). is a Gaussian (non centered if a ~ 0) measure with the same covariance than ~c. Let (en) C E' (orthonormal in H). It suffices to prove the same result for 03B8( ) and 03B8(T03B1( )). But B(Ta(u)) is the product of Gaussian measures on IR with variances one and mean en (a). Therefore it suffices to apply the result of the previous paragraph. Now we shall consider the case where F|H is not nuclear.
We know that in any case F|H is Hilbert-Schmidt.
* Suppose at first that rank (F) is finite.  We have seen the affine case. Now we may give the result for the general case announced in the beginning.  03C9, t Ttw and t -T-1t03C9 are strongly continuously differentiable, the interpolation will be said to be "smooth".
Note that 03C6(1-s)t,t, s E [0,1] is, for t fixed, an interpolation of Tt and naturally (Tt)tE(o,y is an interpolation of T1 : 03C6s,t is a "two-parameter" interpolation of T. dr. S We consider these equations as oDE in Banach space (the first in t with s fixed ; the second in s for t fixed) , we have existence and unicity of solutions with 03C6s,s (03C9) = 03C9, 03C8t,t (03C9) = 03C9 and 03C6s,t o 0 3 C 8 s , t (03C9) = 03C9.
Consequently, ~03C6s,t and ~03C8s,t restricted to H are invertible, and by Ramer's theorem: and P are equivalent. We can obtain another formula for the Radon-Nikodym density using the relation :