On the continuity criteria for Markov processes

We give some criteria and sufficient conditions for the continuity of Markov processes. For example, let $E$ be a locally compact separable metric space and $X$ be a right continuous Markov process on $E$. Suppose the resolvent of $X$ is absolutely continuous in respect to a Radon measure $\mu$, and our condition (B) is fulfilled. If the assertion (10) is valid for any continuous functions $f$ and $g$ with disjoint compact supports, then the process $X$ is continuous almost surely (see Theorem 4). A special case of this result may be found in [1].