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We establish a general result on the existence and uniqueness of a quasi-stationary distribution ₃ of a strongly Feller Markov process (Xₓ, t 0) killed when it exits a domain 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.
Guillin et al. (Thu,) studied this question.