# Heat trace asymptotics on equiregular sub-Riemannian manifolds

## TL;DR

This paper derives detailed short-time asymptotic expansions of the heat trace for a class of sub-Laplacians on compact equiregular sub-Riemannian manifolds, using probabilistic methods and Malliavin calculus.

## Contribution

It provides a comprehensive asymptotic expansion of the heat trace for sub-Laplacians with respect to any smooth measure, highlighting spectral geometric implications for Popp's measure.

## Key findings

- Asymptotic expansion of heat trace up to any order
- Probabilistic proof using Malliavin calculus
- Results applicable to any smooth measure

## Abstract

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use S. Watanabe's distributional Malliavin calculus.