Kolmogorov-Fokker-Planck operators in dimension two: heat kernel and curvature

TL;DR

This paper studies a class of hypoelliptic Kolmogorov-Fokker-Planck operators in two dimensions, explicitly computing heat kernel asymptotics and linking them to curvature-like invariants of an associated control problem, revealing geometric insights.

Contribution

It provides the first explicit computation of heat kernel asymptotics for non-homogeneous Hörmander operators without sub-Riemannian structure, connecting analysis and geometry.

Findings

Explicit first coefficient of small-time heat kernel asymptotics

Interpretation of coefficients as curvature-like invariants

Example of geometric interpretation for non-homogeneous operators

Abstract

We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous H\"ormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the H\"ormander condition.