Hypocoercive estimates on foliations and velocity spherical Brownian motion

TL;DR

This paper extends hypocoercive estimates for hypoelliptic operators on Riemannian foliations, specifically analyzing the velocity spherical Brownian motion, and demonstrates convergence to equilibrium under geometric conditions.

Contribution

It develops a generalized $ abla$-calculus for hypoelliptic operators on non-totally geodesic foliations and applies it to prove hypocoercivity for velocity spherical Brownian motion.

Findings

Established hypocoercive estimates for a broad class of Kolmogorov operators.

Proved convergence to equilibrium in $H^1$ and $L^2$ for velocity spherical Brownian motion.

Identified geometric conditions ensuring hypocoercivity and convergence.

Abstract

By further developing the generalized $\Gamma$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic H\"ormander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.