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I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined.

a) What would be exactly the definition?

b) What reference could be helpful?

EDIT: From what I've read From "Applied Probability and Queues(pg. 198-200)"(Asmussen) I understand that $(X(t))$ is Harris Ergodic if it has a regeneration set (a generalisation of a positive recurrent state) and an invariant distribution.

Is it true that if an embedded chain from $(X(t)) $ is Harris ergodic then $(X(t))$ itself is Harris Ergodic?

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  • $\begingroup$ Did you check Meyn, Tweedie, Stability of Markovian processes II: Continuous-time processes and sampled chains? $\endgroup$ – Joris Bierkens Mar 21 '16 at 18:46
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A continuous time Markov process $X$ is Harris recurrent provided there exists a $\sigma$-finite measure $\mu$ on the state space $S$ of $X$ such that for all Borel sets $B\subset S$, if $\mu(B)>0$ then $\Bbb P^x(\int_0^\infty 1_B(X_t)\,dt=\infty)=1$ for all $x\in S$. See Mesure invariante sur les classes récurrentes des processus de Markov by Azéma, Kaplan-Duflo, and Revuz, D. in Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967) 157–181. They use the fact that an associated embedded chain is Harris recurrent in the discrete time sense.

Under some minimal regularity hypotheses, Kaspi and Mandelbaum, in the paper "On Harris recurrence in continuous time" [Math. Oper. Res. 19 (1994) 211–222] prove a characterization of Harris recurrence in terms of hitting times.

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