Cartan connections for stochastic developments on sub-Riemannian manifolds

TL;DR

This paper introduces a novel approach to constructing sub-Riemannian diffusions using Cartan connections, extending stochastic development concepts from Riemannian to sub-Riemannian geometry.

Contribution

It develops a framework for stochastic development on sub-Riemannian manifolds via Cartan connections, including conditions for their existence and explicit constructions.

Findings

Derived a general expression for the generator of the stochastic process.

Provided a necessary and sufficient condition for the existence of suitable Cartan connections.

Illustrated the construction for free sub-Riemannian structures with two generators.

Abstract

Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.