By Ali S. Üstünel

This publication offers the foundation of the probabilistic sensible research on Wiener house, constructed over the last decade. the topic has advanced significantly in recent times thr- ough its hyperlinks with QFT and the impression of Stochastic Calcu- lus of adaptations of P. Malliavin. even if the latter bargains primarily with the regularity of the legislation of random varia- bles outlined at the Wiener house, the publication makes a speciality of particularly diverse topics, i.e. independence, Ramer's theorem, and so forth. First yr graduate point in practical research and conception of stochastic strategies is needed (stochastic integration with admire to Brownian movement, Ito formulation etc). it may be taught as a 1-semester direction because it is, or in 2 semesters including preliminaries from the idea of stochastic methods it's a basic creation to Malliavin calculus!

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Dys-~ 1 f 0 ih(x,)12ds. Y,] = Z~ = Z~ = E[Eo[ltlyt]E~ = 9Yt], hence E[f(xt)lYt ] = E~ E~ [ZtlYt] If we want to study the smoothness of the measure f ~ E[Y(z,)lYd, then from the above formula, we see that it is sufficient to study the smoothness of f ,--, E~ The reason for the use of p0 is that w and (yt;t E [0, 1]) are two independent Brownian motions under p0 (this follows directly from Paul L~vy's theorem of the characterization of the Brownian motion). After this preliminaries, we can prove the following Theorem Suppose that the map f ~ f(xt) from S ( R d) into D has a continuous extension as a map from S ' ( R d) into D'.

Let us give another result important for the applications: QED P r o p o s i t i o n 3 Let F be in some LP(p) with p > 1 and suppose that the distributional derivative V F of F, is in some Lr(p, H), (1 < r). Then F belongs to D~^p,1. P r o o f i Without loss of generality, we can assume that r <_ p. Let (ei;i r N) be a complete, orthonormal basis of the Cameron-Martin space H. Denote by Vn the sigma-field generated by 5el,... , 6en, and by 7rn the ortohogonal projection of H onto the subspace spanned by e l , .

Proofi As explained above, it is sufficient to prove that the (random) measure has a density in S(Rd). Let /:y be the Ornstein-Uhlenbeck operator on the space of the Brownian motion (yt;t E [0, 1]). ~z, e N LP . P P 51 9 Hence Zt(w, y) C D(w, y), where D(w, y) denotes the space of test functions defined on the product Wiener space with respect to the laws of w and y. 9 The second point is that the operator E~ is a continuous mapping from D~,k(w ,y) into D~ since s commutes with E ~ (for any p_> 1 , k e Z ) .