On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals
type: Item
instance of: scholarly article
title: On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals
zbMATH Open document ID: 0638.60067
author: Hideaki Uemura
published in: Journal of Mathematics of Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: Let (M,g) be a d-dimensional Riemannian manifold, \(\Delta_ M\) be the Laplace-Beltrami operator on M, and p(t,x,y) the fundamental solution of the heat equation \(\partial u/\partial t=2^{-1} \Delta_ Mu\) with respect to the Riemannian volume. The Minakshisundaram-Pleijel expansion \[ (1)\quad p(\epsilon^ 2,x,x)\quad \sim \quad \epsilon^ d(c_ 0+c_ 2\epsilon^ 2+...)\quad (\epsilon \downarrow 0) \] is studied by probabilistic methods. Let \(X^{\epsilon}_ t\) be a \((\epsilon^ 2/2)\Delta_ M\)-diffusion \((\epsilon >0)\). Then \(p(\epsilon^ 2,x,x)\) is expressed as \(E[\delta_ x(X^{\epsilon}_ 1)]\) which can be justified by the Malliavin calculus. Appealing to the asymptotic expansions for generalized Wiener functionals due to \textit{S. Watanabe} [Lectures on stochastic differential equations and Malliavin calculus (1984; Zbl 0546.60054)], we know that \(p(\epsilon^ 2,x,x)\) has the asymptotic expansion (1) and its coefficient \(c_ i\) is given in terms of the curvature and the expectation of multiple Wiener integrals conditioned by \(w(1)=y\), \(w\in W^ d_ 0\) (d-dimensional Wiener space). This conditional expectation can be calculated explicitly by computations of moments of Gaussian processes and brings the principle of ``pairwise contractions''. Thus we can determine the possible types of monomials in components of curvature and its derivatives and see the universality of the coefficients of these monomials.
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 4040975
zbMATH Keywords: heat equation
zbMATH Keywords: fundamental solution
zbMATH Keywords: multiple Wiener integral
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: asymptotic expansions for generalized Wiener functionals
zbMATH Keywords: moments of Gaussian processes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520656
OpenAlex ID: W1597517751
DOI: 10.1215/KJM/1250520656
description: scientific article; zbMATH DE number 4040975