Uniform, localized asymptotics for sub-Riemannian heat kernels, their logarithmic derivatives, and associated diffusion bridges

TL;DR

This paper develops localized small-time asymptotics for sub-Riemannian heat kernels and their derivatives, providing uniform bounds, complete expansions, and insights into the structure of the cut locus and diffusion bridges.

Contribution

It introduces a method to analyze heat kernel asymptotics locally on incomplete manifolds, extending to derivatives and logarithmic derivatives, and characterizes the cut locus via the log-Hessian behavior.

Findings

Uniform heat kernel bounds on compact sets

Complete asymptotic expansions for heat kernels and derivatives

Characterization of the non-abnormal cut locus through log-Hessian behavior

Abstract

We show that the small-time asymptotics of the sub-Riemannian heat kernel, its derivatives, and its logarithmic derivatives can be localized, allowing them to be studied even on incomplete manifolds, under essentially optimal conditions on the distance to infinity. Continuing, away from abnormal minimizers, we show that the asymptotics are closely connected to the structure of the minimizing geodesics between the two relevant points (which is non-trivial on the cut locus). This gives uniform heat kernel bounds on compacts, and also allows a complete expansion of the heat kernel, and its derivatives, in a wide variety of cases. The method extends naturally to logarithmic derivatives of the heat kernel, where we again get uniform bounds on compacts and a more precise expansion for any particular pair of points, in most cases. In particular, we determine the measure giving the law of large numbers for the corresponding diffusion bridge, and the leading terms of the logarithmic derivatives are given by the cumulants of geometrically natural random variables with respect to this measure. One consequence is that the non-abnormal cut locus is characterized by the behavior of the log-Hessian of the heat kernel.