Geodesic
On the continuity criteria for Markov processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 142-149

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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]. 
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     author = {M. G. \v{S}ur},
     title = {On the continuity criteria for {Markov} processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     volume = {25},
     number = {1},
     year = {1980},
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M. G. Šur. On the continuity criteria for Markov processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 142-149. http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a12/