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Consider a CT Markov Process $X=(X_t)_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting distribution and viceversa?

Any useful reference will be much appreciated.

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For Markov chains, a very useful condition is Harris recurrence, see https://en.wikipedia.org/wiki/Harris_chain. This has been generalized to continuous time, see
https://www.jstor.org/stable/3690386?seq=1#metadata_info_tab_contents

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