For a discrete time irreducible + aperiodic markov chain (with general state space X) the geometric ergodicity criteria implies exponential return times to compact sets. More preciesly: The geometric ergodicity criteria is given as $$ \exists c,d>0 :\quad LV(x)\leq -cV(x)+d1_C,\; \forall x\in X$$ where $L$ is the generator corresponding to the markov chain and $C$ is a compact set. This implies that there is $c_1>0$ s.t. $\sup_{x\in C}\mathbb{E}_xe^{c_1\tau_C}<\infty$, where $\tau_C$ is the first time the markov chain arrives in $C$. ( see meyn & tweedies book here )
QUESTION: Is there an equivalent result for a markov process in continuous time? I am mostly interested in a reference for this.