Heat Kernel Comparison on Alexandrov Spaces with Curvature Bounded Below

Abstract

In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with lower curvature bound in the sense of Alexandrov and with sufficiently fast asymptotic decay of the volume of small geodesic balls. As corollaries we recover Varadhan's short time asymptotic formula for the heat kernel (1967) and Cheng's eigenvalue comparison theorem (1975). Finally, we derive an integral inequality for the distance process of a Brownian Motion on (X,d) resembling earlier results in the smooth setting by Debiard, Geavau and Mazet (1975).