  We are concerned with stochastic processes on surfaces in three-dimensional
contact sub-Riemannian manifolds. Employing 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 ${\rm SU}(2)$ and ${\rm SL}(2,\mathbb{R})$ equipped with the
standard sub-Riemannian contact structures as model cases for this setting. Our
techniques further allow us to derive an expression for an intrinsic Gaussian
curvature of a surface in a general three-dimensional contact sub-Riemannian
manifold.
