Curvature terms in small time heat kernel expansion for a model class of hypoelliptic H\"{o}rmander operators

TL;DR

This paper analyzes the small-time behavior of heat kernels for a class of hypoelliptic operators, linking the expansion coefficients to geometric invariants like divergence and curvature, enhancing understanding of their geometric structure.

Contribution

It provides a geometric characterization of the small-time heat kernel expansion coefficients for hypoelliptic H"{o}rmander operators, connecting them to divergence and curvature invariants.

Findings

Derived explicit small-time heat kernel expansion on the diagonal.

Linked expansion coefficients to divergence of drift and curvature invariants.

Enhanced geometric understanding of hypoelliptic operator heat kernels.

Abstract

We consider the heat equation associated with a class of second order hypoelliptic H\"{o}rmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the drift field and the curvature-like invariants of the optimal control problem associated with the diffusion operator.