Quasi-stationary distribution for kinetic SDEs with low regularity coefficients

TL;DR

This paper establishes the existence and uniqueness of quasi-stationary distributions for kinetic SDEs with low regularity coefficients, using Harnack inequalities and Lyapunov conditions in specific domains.

Contribution

It introduces a novel approach to prove quasi-stationary distributions for kinetic SDEs with irregular coefficients, extending previous results to more general settings.

Findings

Proved a Harnack inequality for solutions of low-regularity kinetic SDEs.

Established existence and uniqueness of quasi-stationary distributions under Lyapunov conditions.

Identified specific settings where the Lyapunov condition holds, including bounded position domains and Langevin processes.

Abstract

We consider kinetic SDEs with low regularity coefficients in the setting recently introduced in [6]. For the solutions to such equations, we first prove a Harnack inequality. Using the abstract approach of [5], this inequality then allows us to prove, under a Lyapunov condition, the existence and uniqueness (in a suitable class of measures) of a quasi-stationary distribution in cylindrical domains of the phase space. We finally exhibit two settings in which the Lyapunov condition holds: general kinetic SDEs in domains which are bounded in position, and Langevin processes with a non-conservative force and a suitable growth condition on the force.