Hypocoercive estimates on foliations and velocity spherical Brownian motion

Fabrice Baudoin; Camille Tardif

doi:10.3934/krm.2018001

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2018, Volume 11, Issue 2: 1-23. Doi: 10.3934/krm.2018001

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Hypocoercive estimates on foliations and velocity spherical Brownian motion

Fabrice Baudoin^1, , and
Camille Tardif^2,

1.
Department of Mathematics, University of Connecticut, 341 Mansfield Road Storrs, CT 06269-1009, USA
2.
LPMA, Université Pierre et Marie Curie, 4, Place Jussieu 75005 Paris, France

* Corresponding author: Fabrice Baudoin
* Corresponding author: Fabrice Baudoin

Received: April 2016

Revised: April 2017

Published: April 2018

The first author is supported in part by Grant NSF-DMS 15-11-328

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Abstract

By further developing the generalized $Γ$-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 Hörmander'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 $L^2$.

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Mathematics Subject Classification: Primary:60J60, 58J65, 35H10, 35B40;Secondary:55R25.

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