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 Uspekhi Mat. Nauk, 1973, Volume 28, Issue 2(170), Pages 35–64 (Mi umn4861)

Regular Markov processes

E. B. Dynkin

Abstract: This article is concerned with the foundations of the theory of Markov processes. We introduce the concepts of a regular Markov process and the class of such processes. We show that regular processes possess a number of good properties (strong Markov character, continuity on the right of excessive functions along almost all trajectories, and so on). A class of regular Markov processes is constructed by means of an arbitrary transition function (regular re-construction of the canonical class). We also prove a uniqueness theorem. We diverge from tradition in three respects: a) we investigate processes on an arbitrary random time interval; b) all definitions and results are formulated in terms of measurable structures without the use of topology (except for the topology of the real line); c) our main objects of study are non-homogeneous processes (homogeneous ones are discussed as an important special case). In consequence of a), the theory is highly symmetrical: there is no longer disparity between the birth time $\alpha$ of the process, which is usually fixed, and the terminal time $\beta$, which is considered random. Principle b) does not prevent us from introducing, when necessary, various topologies in the state space (as systems of coordinates are introduced in geometry). However, it is required that the final statements should be invariant with respect to the choice of such a topology. Finally, the main gain from c) is simplification of the theory: discarding the “burden of homogeneity” we can use constructions which, generally speaking, destroy this homogeneity. Similar questions have been considered (for the homogeneous case) by Knight [8], Doob [2], [3] and other authors.

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English version:
Russian Mathematical Surveys, 1973, 28:2, 33–64

Bibliographic databases:

MSC: 60Jxx, 60Gxx

Citation: E. B. Dynkin, “Regular Markov processes”, Uspekhi Mat. Nauk, 28:2(170) (1973), 35–64; Russian Math. Surveys, 28:2 (1973), 33–64

Citation in format AMSBIB
\Bibitem{Dyn73} \by E.~B.~Dynkin \paper Regular Markov processes \jour Uspekhi Mat. Nauk \yr 1973 \vol 28 \issue 2(170) \pages 35--64 \mathnet{http://mi.mathnet.ru/umn4861} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=400410} \zmath{https://zbmath.org/?q=an:0334.60031|0385.60059} \transl \jour Russian Math. Surveys \yr 1973 \vol 28 \issue 2 \pages 33--64 \crossref{https://doi.org/10.1070/RM1973v028n02ABEH001529} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. M. G. Shur, “Ob approksimatsii additivnykh funktsionalov”, UMN, 29:6(180) (1974), 183–184
2. E. B. Dynkin, “Markov representations of stochastic systems”, Russian Math. Surveys, 30:1 (1975), 65–104
3. E.B Dynkin, “Additive functionals of several time-reversible Markov processes”, Journal of Functional Analysis, 42:1 (1981), 64
4. E.B Dynkin, “Green's and Dirichlet spaces associated with fine Markov processes”, Journal of Functional Analysis, 47:3 (1982), 381
5. È. B. Vinberg, S. E. Kuznetsov, “Evgenii (Eugene) Borisovich Dynkin (obituary)”, Russian Math. Surveys, 71:2 (2016), 345–371
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