On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

TL;DR

This paper characterizes the Onsager-Machlup functional for hypoelliptic Brownian motion on the Heisenberg group, providing insights into its trajectory properties without relying on traditional Riemannian geometric tools.

Contribution

It introduces a novel analysis of the Onsager-Machlup functional in the sub-Riemannian setting, specifically for the Heisenberg group, bypassing classical differential geometric methods.

Findings

Derived the Onsager-Machlup functional for hypoelliptic Brownian motion

Analyzed trajectory properties including horizontal continuous curves

Provided a framework applicable in sub-Riemannian geometry

Abstract

Onsager-Machlup functionals are used to describe the dynamics of a continuous stochastic process. For a stochastic process taking values in a Riemannian manifold, they have been studied extensively. We describe the Onsager-Machlup functional with respect to the sup norm for a hypoelliptic Brownian motion on a Heisenberg group. Unlike in the Riemannian case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not easily available in the sub-Riemannian setting. In addition, we study fine properties of trajectories of the hypoelliptic Brownian motion, including a new notion of horizontal continuous curves.