Stochastic analysis on sub-Riemannian manifolds with transverse symmetries

TL;DR

This paper develops a stochastic representation for the heat semigroup derivative on sub-Riemannian manifolds with transverse symmetries, leading to new gradient bounds and insights even for the Heisenberg group.

Contribution

It introduces a geometrically meaningful stochastic representation of the heat semigroup derivative on sub-Riemannian manifolds with transverse symmetries, extending understanding of hypoelliptic heat kernels.

Findings

Derived a stochastic representation for the heat semigroup derivative.

Established new hypoelliptic heat semigroup gradient bounds.

Results apply to the Heisenberg group, a key example of such manifolds.

Abstract

We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions. The results are new even in the case of the Heisenberg group which is the simplest example of a sub-Riemannian manifold with transverse symmetries.