Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus

TL;DR

This paper derives a detailed short-time asymptotic expansion of hypoelliptic heat kernels on Euclidean spaces and manifolds, especially in the complex cut locus case, using probabilistic methods and advanced stochastic calculus.

Contribution

It provides the first comprehensive asymptotic expansion of hypoelliptic heat kernels at the cut locus, employing Malliavin calculus and rough path theory.

Findings

Asymptotic expansion valid up to any order in the cut locus case.

Probabilistic approach using stochastic differential equations.

Application of Malliavin calculus and rough path theory to hypoelliptic heat kernels.

Abstract

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.