Kinetic Brownian motion on Riemannian manifolds

TL;DR

This paper introduces a family of hypoelliptic diffusion processes called kinetic Brownian motions on Riemannian manifolds, interpolating between geodesic and Brownian motions, and analyzes their long-term behavior and Poisson boundary.

Contribution

It provides a new stochastic model for geodesic flow perturbations and thoroughly studies its asymptotic properties on rotationally invariant manifolds.

Findings

Kinetic Brownian motions interpolate between geodesic and Brownian motions.

Complete description of the Poisson boundary on rotationally invariant manifolds.

Quantitative analysis of the long-time behavior of these processes.

Abstract

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $\sigma$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$. We show, both qualitively and quantitatively, that the laws of these $\mathcal M$-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter $\sigma$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $\sigma$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.