Dirichlet sub-Laplacians on homogeneous Carnot groups: spectral properties, asymptotics, and heat content

TL;DR

This paper investigates the spectral properties of Dirichlet sub-Laplacians on homogeneous Carnot groups, establishing their spectral gap, and explores applications to hypoelliptic Brownian motion and heat content asymptotics.

Contribution

It provides new results on the spectral gap and eigenvalues of sub-Laplacians in Carnot groups, with applications to stochastic processes and heat analysis.

Findings

Operators have a pure point spectrum

Existence of a spectral gap proven

Results applied to hypoelliptic Brownian motion and heat content asymptotics

Abstract

We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.