Résumé : In this talk I will connect via the random walks point of view, the geodesics and the Laplacian in a sub-Riemannian manifold $M$. This problem is not trivial even in the Riemannian context and passes through the definition of a volume. We define two type of Laplacians : the macroscopic one as the divergence of the horizontal gradient once a volume in the ambient space is fixed and the microscopic one as the operator associated with a geodesic random walk. This second definition requires to fix a probability density on the cylinder of initial conditions of covectors in $T^\ast M_q$, where $q$ is the starting point. This cylinder parametrizes the different geodesics. We study under which conditions these two operator coincide. We will see that the result strongly depend on the type of sub-Riemannian structure. The main purpose is to understand which is the most natural diffusion equation that should be used in an algorithm of image reconstruction.

Lieu : Salle 113-115, Bâtiment 425