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

ugo v. boscain, Andrei Agrachev, Jean-Paul Gauthier, Francesco Rossi
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.
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