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Zbl 0664.46028
Feyel, D.; de la Pradelle, Arnaud
Espaces de Sobolev gaussiens. (Gaussian Sobolev spaces). (French)
[J] Ann. Inst. Fourier 39, No.4, 875-908 (1989). ISSN 0373-0956; ISSN 1777-5310/e

Let $\mu$ be a gaussian measure on a locally convex linear space E. We give a new point of view on the first Sobolev space W(E,$\mu)$ built on E with respect to $\mu$. The differential f' of $f\in W(E,\mu)$ is a function of two variables (x,y)$\in E\times E$, which is ``quasi-linear'' in the second variable. \par The differential of a stochastic integral is a stochastic integral on $E\times E$ with respect to $\mu$ $\otimes \mu.$ \par The natural ``gaussian procapacity'' is a true capacity if E is a Banach or a Fréchet space or the weak dual of a separable Fréchet nuclear space. \par Any $f\in W(E,\mu)$ is equal $\mu$-almost everywhere to a quasi- continuous function g, moreover any such g has for any Cameron-Martin direction the absolute continuity property in almost every line parallel to this direction.
[D.Feyel]

MSC 2000:: *46E35 Sobolev spaces and generalizations
46G12 Measures and integration on abstract linear spaces
60H05 Stochastic integrals

Keywords: Gaussian measure on a locally convex linear space; Sobolev space; differential of a stochastic integral; Gaussian procapacity; weak dual of a separable Frechet nuclear space; Cameron-Martin direction

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