Geodesic
On the starting points of wanderings of Markov processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 490-493

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Let $x_t$ be a Markov process on $E$ and $D$ be a subset of $E$. We will call a wandering any connected component of the set $\{t\colon x_t\in D\}$. Denote by $\overline x_t$ the process obtained from $x_t$ by killing at the first exit time out of $D$. It is proved that, under some conditions, with probability 1, every wandering starts at a point of the Martin boundary corresponding to $\overline x_t$ (i.e. the limit in (3) exists). 
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     author = {E. B. Dynkin and A. A. Yushkevich},
     title = {On the starting points of wanderings of {Markov} processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {490--493},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a7/}
}
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E. B. Dynkin; A. A. Yushkevich. On the starting points of wanderings of Markov processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 490-493. http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a7/