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arXiv:0806.0734v2[math.AP]13Jun2008 The intrinsic hypoelliptic Laplacian and its heat kernel
 

Summary: arXiv:0806.0734v2[math.AP]13Jun2008
The intrinsic hypoelliptic Laplacian and its heat kernel
on unimodular Lie groups
Andrei Agrachev
SISSA, via Beirut 2-4, 34014 Trieste, Italy - agrachev@sissa.it
Ugo Boscain
LE2i, CNRS UMR5158, Universit´e de Bourgogne, 9, avenue Alain Savary - BP 47870, 21078 Dijon CEDEX, France -
boscain@sissa.it
Jean-Paul Gauthier
Laboratoire LSIS, Universit´e de Toulon, France - gauthier@univ-tln.fr
Francesco Rossi
SISSA, via Beirut 2-4, 34014 Trieste, Italy - rossifr@sissa.it
June 16, 2008
Abstract
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with con-
stant 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

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics