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Asymptotic expansion for transport processes in semi-Markov media. (English. Russian original) Zbl 1253.60094

Theory Probab. Math. Stat. 83, 127-134 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 106-112 (2010).
There are many approaches dealing with the asymptotic expansions for perturbed semi-Markov random evolutions (e.g., [S. Albeverio, V. S. Korolyuk and I. V. Samoilenko, Stochastics 81, No. 5, 477–502 (2009; Zbl 1179.60058)]). This paper finds solutions of singularly perturbed equations of semi-Markov random evolutions by reducing the semi-Markov process to an equivalent Markov process consisting of three known processes. However, the resulting Markov process has a complicated continuous phase space. Using the results by V. S. Korolyuk and A. F. Turbin [Markov renewal processes in problems of systems reliability (Russian). Kiev: “Naukova Dumka” (1982; Zbl 0508.60073)], the authors apply the method of asymptotic expansion for Markov random evolutions.

MSC:

60K15 Markov renewal processes, semi-Markov processes
35C20 Asymptotic expansions of solutions to PDEs
35R60 PDEs with randomness, stochastic partial differential equations
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