Dirichlet spaces and strong Markov processes

Fukushima, Masatoshi

doi:10.1090/S0002-9947-1971-0295435-0

Dirichlet spaces and strong Markov processes
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Trans. Amer. Math. Soc. 162 (1971), 185-224 Request permission

Abstract:

We show that there exists a suitable strong Markov process on the underlying space of each regular Dirichlet space. Potential theoretic concepts due to A. Beurling and J. Deny are then described in terms of the associated strong Markov process. The proof is carried out by developing potential theory for Dirichlet spaces and symmetric Ray processes and by using a method of transformation of underlying spaces.

References

A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208–215. MR 106365, DOI 10.1073/pnas.45.2.208
R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
Jacques Deny, Théorie de la capacité dans les espaces fonctionnels, Sém. Théorie Potentiel, dirigé par M. Brelot, G. Choquet et J. Deny, 1964/65, Exp. 1, Secrétariat mathématique, Paris, 1964/1965, pp. 13 (French). MR 0190367
Potential theory, Centro Internazionale Matematico Estivo (C.I.M.E.), Edizioni Cremonese, Rome, 1970. I Ciclo, Stresa, 2–10 Luglio 1969; Coordinatore: M. Brelot. MR 0271378
J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 21 (1945), 305–370 (1955) (French). MR 74787
J. L. Doob, Applications to analysis of a topological definition of smallness of a set, Bull. Amer. Math. Soc. 72 (1966), 579–600. MR 203665, DOI 10.1090/S0002-9904-1966-11533-1
E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
Masatoshi Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka Math. J. 4 (1967), 183–215. MR 231444
Masatoshi Fukushima, On Dirichlet spaces and Dirichlet rings, Proc. Japan Acad. 45 (1969), 433–436. MR 253395
Masatoshi Fukushima, Regular representations of Dirichlet spaces, Trans. Amer. Math. Soc. 155 (1971), 455–473. MR 281256, DOI 10.1090/S0002-9947-1971-0281256-1
G. A. Hunt, Markoff processes and potentials. I, II, Illinois J. Math. 1 (1957), 44–93, 316–369. MR 91349
Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
Hiroshi Kunita and Takesi Watanabe, Some theorems concerning resolvents over locally compact spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 131–164. MR 0214148
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
Paul-André Meyer, Processus de Markov, Lecture Notes in Mathematics, No. 26, Springer-Verlag, Berlin-New York, 1967 (French). MR 0219136
Mitsuru Nakai, Algebraic criterion on quasiconformal equivalence of Riemann surfaces, Nagoya Math. J. 16 (1960), 157–184. MR 110801
Daniel Ray, Resolvents, transition functions, and strongly Markovian processes, Ann. of Math. (2) 70 (1959), 43–72. MR 107302, DOI 10.2307/1969891