Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds (Q2077327)

Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds are considered. A new second-order partial differential operator on such surfaces arising as the limit of Laplace-Beltrami operators is obtained. The stochastic process associated with this operator moves along the characteristic foliation induced on the surface by the contact distribution is studied. It is proved that for this stochastic process, elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. The illustrations of these results on quadric surfaces in the Heisenberg group and on canonical surfaces in the Lie groups \(\mathrm{SU}(2)\) and \(\mathrm{SL}(2,\mathbb{R})\) equipped with the standard sub-Riemannian contact structures are presented. As an application of the results and constructions of the paper, an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold is deduced.