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January, 1987 Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels
Shinzo Watanabe
Ann. Probab. 15(1): 1-39 (January, 1987). DOI: 10.1214/aop/1176992255

Abstract

An analysis of Wiener functionals is studied as a kind of Schwartz distribution theory on Wiener space. For this, we introduce, besides ordinary $L_p$-spaces of Wiener functionals, Sobolev-type spaces of (generalized) Wiener functionals. Any Schwartz distribution on $\mathbf{R}^d$ is pulled back to a generalized Wiener functional by a $d$-dimensional Wiener map which is smooth and nondegenerate in the sense of Malliavin. As applications, we construct a heat kernel (i.e., the fundamental solution of a heat equation) by a generalized expectation of the Dirac delta function pulled back by an Ito map, i.e., a Wiener map obtained by solving Ito's stochastic differential equations. Short-time asymptotics of heat kernels are studied through the asymptotics, in terms of Sobolev norms, of the generalized Wiener functional under the expectation.

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Shinzo Watanabe. "Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels." Ann. Probab. 15 (1) 1 - 39, January, 1987. https://doi.org/10.1214/aop/1176992255

Information

Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0633.60077
MathSciNet: MR877589
Digital Object Identifier: 10.1214/aop/1176992255

Subjects:
Primary: 60H10
Secondary: 28C20 , 35K05

Keywords: asymptotic expansion of Wiener functionals , generalized Wiener functionals , Heat kernels , pull-back of Schwartz distributions , short-time asymptotics , Sobolev spaces of Wiener functionals

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 1 • January, 1987
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