Strong Short Time Asymptotics and Convolution Approximation of the Heat Kernel

Abstract

We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of $M$, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus.

28 pages; minor changes; some references added

References in corpus (3)

• Initial value problems for wave equations on manifolds
• A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel
• Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac

Cited by in corpus (7)

• Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants
• Analytic semigroups approaching a Schrödinger group on real foliated metric manifolds
• Heat Kernels as Path Integrals
• A geometric Laplace method
• Abstract McKean-Vlasov and HJB equations, their fractional versions and related forward-backward systems on Riemannian manifolds
• Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus
• Localized bounds on log-derivatives of the heat kernel on incomplete Riemannian manifolds