Quasi-stationary distribution for strongly Feller Markov processes by Lyapunov functions and applications to hypoelliptic Hamiltonian systems

Abstract

We establish a general result on the existence and uniqueness of a quasi-stationary distribution ${\mu }_{\mathcal{D}}$ of a strongly Feller Markov process $(X_t,t\ge 0)$ killed when it exits a domain $\mathcal{D}$, under some Lyapunov function condition. Our result covers the case of hypoelliptic damped Hamiltonian systems. Our method is based on a characterization of the essential spectral radius by means of Lyapunov functions and measures of noncompactness.