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1. Linear p e r t u r b a t i o n s of H - v a l u e d fields. Let X be a locally convex space equippe d with a centered Gaussian measure 7. ~ the closure of X* in L2(,/) (which is known to coincide with the space of all '/-measurable linear functionals on X, see [10, Chap. 3; 16]). We say that two Gaussian spaces (X1, ,/1) and (X2, '/2) are linearly isomorphic if there is a ,/Fmeasurable linear mapping J : X1 --* X2 such that ,/1 o j - 1 = '/2 and J : H(71 ) --* H(72 ) is an isometry of Hilbert spaces.

Relat. Fields, 93, 91-136 (1992). 47. A. S. and M. Zakai, "Analyse de rotations al~atoires sur l'espace de Wiener," C. R. Acad. Sci. S~r. I, 319, 1069-1073 (1994). 48. A. S. 0stfinel and M. Zakai, "Random rotations of the Wiener path," Probab. Theor. Relat. Fields, 103, No. 3, 409-430 (1995). 49. N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nanka, Moscow (1985). 50. M. Zakai and O. " Ann. , 20, No. 3, 1436-1440 (1992). 51. W. Ziemer, Weakly Differentiable Functions, Springer (1989).

1. Assume that # is a probability measure on a locally convex space X , and for a measurable linear mapping T from X into another locally convex space Y , let B T be the sub-a-algebra {T-I(A), A 6 By}. /fIt is differentiable along'h ~ X , then ItT is differentiable along Th, and P r o o f . 1) follows, since r o T generates B T. 2. Assume that a probability measure It on a locally convex space X is differentiable in directions . h l , . . , h n , and for some c > O, one has f exp(cl~h,(x)l)It(d~) < (3C.

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Dynamical systems generated by sobolev class vector fields in finite and infinite dimensions


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