Open Access
Translator Disclaimer
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


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.


Download Citation

Shinzo Watanabe. "Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels." Ann. Probab. 15 (1) 1 - 39, January, 1987.


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

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

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


Vol.15 • No. 1 • January, 1987
Back to Top