On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals
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.
asymptotic expansions for generalized Wiener functionals
scientific article; zbMATH DE number 4040975