Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

Abstract: We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. By considering the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in $\mathsf{SU}(2)$ and $\mathsf{SL}(2,\mathbb{R})$ equipped with the standard sub-Riemannian contact structures as model cases for this setting. This is joint work with Davide Barilari, Ugo Boscain and Daniele Cannarsa.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021