Dirichlet Sub-Laplacians on Homogeneous Carnot Groups: Spectral Properties, Asymptotics, and Heat Content

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.