We establish a general result on the existence and uniqueness of a quasi-stationary distribution μD of a strongly Feller Markov process (Xₜ,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.
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Guillin et al. (2024) studied this question.
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