Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

TL;DR

This paper constructs and analyzes an intrinsic sub-Laplacian on hypersurfaces within contact sub-Riemannian manifolds, demonstrating its properties and behavior near characteristic points through model cases.

Contribution

It introduces a new intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds and studies its properties via Riemannian approximations and model examples.

Findings

The intrinsic sub-Laplacian is the limit of Riemannian Laplacians away from characteristic points.

It is stochastically complete in model cases.

The induced stochastic process almost surely avoids characteristic points.

Abstract

We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace-Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.