In this talk I will introduce the quasi-stationary distribution, the equilibrium of a Markov process conditionned to be inside a domain，and the quasi-ergodic distribution，the limit distribution of the empirical process. These problems come from biological mathematics and statistical mechanics. Our approaches are the formula of the essential spectral radius and generalized Perron-Frobenius theorem. This talk is based on a series of my joint works with Guillin and Nectoux in JEMS, PTRF and Ann Math Toulouse.

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