Short-time asymptotics of heat kernels of hypoelliptic Laplacians on Lie groups

Abstract

This thesis suggests an approach to compute the short-time behaviour of the hypoelliptic heat kernel corresponding to sub-Riemannian structures on unimodular Lie groups of type I, without previously holding a closed form expression for this heat kernel. Our work relies on the use of classical non-commutative harmonic analysis tools, namely the Generalized Fourier Transform and its inverse, combined with the Trotter product formula from the theory of perturbation of semigroups. We illustrate our main results by computing, to our knowledge, a first expression in short-time for the hypoelliptic heat kernel on the Engel and the Cartan groups, for which there exist no closed form expression.