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Denumerable Markov processes and the associated contraction semigroups on l. (English) Zbl 0079.34703


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[1] D. G. Austin, On the existence of the derivative of Markoff transition probability functions.Proc. Nat. Acad. Sci. U.S.A., 41 (1955), 224–226. · Zbl 0068.12503 · doi:10.1073/pnas.41.4.224
[2] K. L. Chung, Some new developments in Markov chains,Trans. Am. Math. Soc., 81 (1956), 195–210. · Zbl 0075.14001 · doi:10.1090/S0002-9947-1956-0075481-4
[3] R. L. Dobrušin, On conditions of regularity of stationary Markov processes with a denumerable number of possible states.Uspehi Matem. Nauk (N. S.), 7 (1952), 185–191.
[4] J. L. Doob, Markoff chains–denumerable case.Trans. Am. Math. Soc., 58 (1945), 455–473. · Zbl 0063.01146
[5] J. L. Doob,Stochastic Processes. New York & London. 1953.
[6] W. Feller, On the integro-differential equations of purely discontinuous Markoff process.Trans. Am. Math. Soc., 48 (1940), 488–515;ibid., 58 (1945), 474. · Zbl 0025.34704 · doi:10.1090/S0002-9947-1940-0002697-3
[7] W. Feller,An Introduction to Probability Theory and Its Applications, I. New York & London, 1950. · Zbl 0039.13201
[8] –, Boundaries induced by non-negative matrices.Trans. Am. Math. Soc., 83 (1956), 19–54. · Zbl 0071.34901 · doi:10.1090/S0002-9947-1956-0090927-3
[9] W. Feller, On boundary conditions for the Kolmogorov differential equations (to appear). · Zbl 0084.35503
[10] E. Hille,Functional Analysis and Semi-Groups. New York, 1948. · Zbl 0033.06501
[11] E. Hille, On the generation of semi-groups and the theory of conjugate functions.Kungl. Fysiografiska Sällskapets i Lund Förhandlingar, 21 (1952), No. 14.
[12] –, Perturbation methods in the study of Kolmogorov’s equations.Proceedings of the International Congress of Mathematicians (1954), Vol. III, 365–376.
[13] A. Jensen,A Distribution Model. Copenhagen, 1954. · Zbl 0055.38005
[14] S. Karlin &J. McGregor, Representation of a class of stochastic processes.Proc. Nat. Acad. Sci. 41 (1955), 387–391. · Zbl 0067.10803 · doi:10.1073/pnas.41.6.387
[15] T. Kato, On the semigroups generated by Kolmogoroff’s differential equations.J. Math. Soc. Japan, 6 (1954), 1–15. · Zbl 0058.10701 · doi:10.2969/jmsj/00610001
[16] D. G. Kendall, Some analytical properties of continuous stationary Markov transition functions.Trans. Am. Math. Soc., 78 (1955), 529–540. · Zbl 0068.12501 · doi:10.1090/S0002-9947-1955-0067401-2
[17] –, Some further pathological examples in the the theory of denumerable Markov processes.Quart. J. of Math. Oxford (Ser. 2), 7 (1956), 39–56. · Zbl 0075.14101 · doi:10.1093/qmath/7.1.39
[18] D. G. Kendall &G. E. H. Reuter, Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators inl.Proceedings of the International Congress of Mathematicians (1954), Vol. III, 377–415.
[19] A. N. Kolmogorov, On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states.Moskov. Gos. Univ. Učenye Zapiski Matematika, 148 (1951), 53–59.
[20] W. Ledermann &G. E. H. Reuter, Spectral theory for the differential equations of simple birth ard death processes.Phil. Trans. Roy. Soc. London (Ser. A), 246 (1954), 321–369. · Zbl 0059.11704 · doi:10.1098/rsta.1954.0001
[21] P. Lévy, Systèmes markoviens et stationnaires; cas dénombrable,Ann. Sci. École-Norm. Sup. (3), 68 (1951), 327–381. · Zbl 0044.33803
[22] G. E. H. Reuter, A note on contraction semigroups.Math. Scand., 3 (1956), 275–280. · Zbl 0067.35301
[23] G. E. H. Reuter &W. Ledermann, On the differential equations for the transition probabilities of Markov processes with enumerably many states.Proc. Camb. Phil. Soc., 49 (1953), 247–262. · Zbl 0053.27202 · doi:10.1017/S0305004100028346
[24] K. Yosida, On the differentiability and the representation of one-parameter semi-groups of linear operators.J. Math. Soc. Japan, 1 (1948), 15–21. · Zbl 0037.35302 · doi:10.2969/jmsj/00110015
[25] –, An operator-theoretical treatment of temporally homogenous MMarkoff process.Ibid.,, 1 (1949), 244–253. · Zbl 0039.35201 · doi:10.2969/jmsj/00130244
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