Any Markov process in a~Borel space has a~transition function
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 389-393
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The most natural way to define «Markov process» $x_t$ is to say that it is a stochastic process with the Markov property. However, in some of the most interesting applications it is possible to consider only the processes with transition function, which is a family $\{p_t^s(x,\Gamma)\}$ of conditional distributions of $x_t$ given $x_s$, satisfying Kolmogorov–Chapman equation $p_t^s p_u^t=p_u^s$, $s$. We prove that the Markov process has the transition function if its state space is universal (i. e. it is isomorphic to a universally measurable subset of a Polish space).
@article{TVP_1980_25_2_a16,
author = {S. E. Kuznecov},
title = {Any {Markov} process in {a~Borel} space has a~transition function},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {389--393},
publisher = {mathdoc},
volume = {25},
number = {2},
year = {1980},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a16/}
}
S. E. Kuznecov. Any Markov process in a~Borel space has a~transition function. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 389-393. http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a16/