The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

Abstract

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace–Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU ( 2 ) , SO ( 3 ) , SL ( 2 ) (with the metrics inherited by the Killing form), and the group SE ( 2 ) of rototranslations of the plane.