Diffusion in small time in incomplete sub-Riemannian manifolds

TL;DR

This paper establishes conditions under which Gaussian upper bounds and small-time asymptotics hold for heat kernels on incomplete sub-Riemannian manifolds, revealing insights into diffusion behavior and measure concentration.

Contribution

It identifies two alternative conditions ensuring Gaussian bounds and asymptotics for heat kernels in incomplete sub-Riemannian settings, extending existing theory.

Findings

Gaussian upper bounds with optimal constants are valid under the conditions.

Small-time logarithmic asymptotics of the heat kernel are derived.

Diffusion bridge measures concentrate near minimal energy paths.

Abstract

For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.