Root
/
Conference Proceedings
/
4th International Conference on Physics and Control (PhysCon 2009)
/
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups
/

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with

constant growth vector using the Poppâ€™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.

constant growth vector using the Poppâ€™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.