Quasi-stationary distribution for kinetic SDEs with low regularity coefficients

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.