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Some new results on transition probability. (English) Zbl 1152.60339

Summary: We study the basic properties of stationary transition probability of Markov processes on a general measurable space \((E, \mathcal E)\), such as the continuity, maximum probability, zero point, positive probability set, standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Lévy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space \((E, \mathcal E)\). Our main tools such as Egoroff’s Theorem, Vitali-Hahn-Saks’s Theorem and the theory of atomic set and well-posedness of measure are also very interesting and fashionable.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60J35 Transition functions, generators and resolvents
47D03 Groups and semigroups of linear operators
47D07 Markov semigroups and applications to diffusion processes
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[1] Doob, J. L.: Stochastic Processes, John Wiley and Sons, 1953 · Zbl 0053.26802
[2] Wang, Z. K.: Theory of stochastic processes, Beijing, Science Press, in Chinese, 1965 · Zbl 0139.43801
[3] Dynkin, E. B.: Markov processes and related problems of analysis, Cambridge University Press, 1982 · Zbl 0498.60067
[4] Yang, X. Q.: The construction theory of denumerable Markov process, John Wiley and Sons, 1990 · Zbl 0788.60088
[5] Wang, Z. K., Yang, X. Q.: Birth and death processes and Markov chains, Berlin, Springer-Verlag, Science Press, 1992 · Zbl 0773.60065
[6] Loève, M.: Probability Theory (I) (II), Berlin, Springer-Verlag, Science Press, 1977
[7] Kingman, J. F. C.: Markov transition probabilities (I). Probability Theory and Related Fields, 7, 248–270 (1967) · Zbl 0183.20101
[8] Kingman, J. F. C.: Markov transition probabilities (II). Probability Theory and Related Fields, 9, 1–9 (1967) · Zbl 0201.50402
[9] Kingman, J. F. C.: Markov transition probabilities (III). Probability Theory and Related Fields, 10, 87–101 (1968) · Zbl 0183.20102
[10] Kingman, J. F. C.: Markov transition probabilities (IV). Probability Theory and Related Fields, 11, 9–17 (1969) · Zbl 0182.22804
[11] Kingman, J. F. C.: Markov transition probabilities (V). Probability Theory and Related Fields, 2, 89–103 (1971) · Zbl 0196.20103
[12] Doob, J. L.: Topics in the theory of Markoff chains. Trans. Amer. Math. Soc., 52, 37–64 (1942) · Zbl 0063.09001
[13] Doob, J. L.: Markoff chains. Denumerable case. Trans. Amer. Math. Soc., 58, 455–473 (1945) · Zbl 0063.01146
[14] Lévy, P.: Systèmes markoviens et stationnaires. Cas dênombrable. Annales Scientifiques de l’êcole Normale Supérieure Sér., 68(3), 327–381 (1951)
[15] Doob, J. L.: State space for Markov chains. Trans. Amer. Math. Soc., 149, 279–305 (1970) · Zbl 0231.60048
[16] Cairoli, M. R.: Une classe de processus de Markov. C. R. Acad. Sc, Paris, Ser. A., 273, 1071–1074 (1971) · Zbl 0238.60049
[17] Wang, Z. K.: Ornstein-Uhlenbeck processes with two parameters. Acta Mathematica Scientia, English Series, 4(1), 1–12 (1984) · Zbl 0549.60049
[18] Wang, Z. K.: The transition probability and prediction on Ornstein-Uhlenbeck processes with two parameters. Chinese Science Bulletin, English Series, 33(1), 5–9 (1988) · Zbl 0664.60055
[19] Xie, Y. Q.: The transition functions of two parameters Markov chains, Master Dissertation of Xiangtan University, (in Chinese), 1–69, 1988
[20] Guo, J. Y., Yang, X. Q.: Three-point transition function for two-parameter Markov chains and their four systems of partial differential equations. Science in China, Series A, 35(7), 806–818 (1992) · Zbl 0758.60071
[21] Zhou, J. W.: The two-parameter regular Markov processes and processes with independent increments. Chinese Journal of Applied Probability and statistics, 8(3), 281–288 (1992) · Zbl 0952.60520
[22] Liu, S. Y.: Density functions of three points transition functions of two-parameter Markov chains. Natural science journal of Xiangtan University, (in Chinese), 15(3), 26–27 (1993) · Zbl 0793.60079
[23] Yang, X. Q., Liu, L. X.: Two-parameter random walk. Acta Mathematica Scientia, Chinese Series, 15(4), 361–367 (1995) · Zbl 0900.60082
[24] Yang, X. Q., Li, Y. Q.: Markov processes with two parameters, in Chinese, Hu’nan Science and Technology Press, Changsha, 1996
[25] Yang, X. Q.: Wide-past homogeneous Markov processes with two parameters and two states three-point transition function. Chinese Science Bulletin, (in Chinese), 41(2), 192–193 (1996)
[26] Li, Y. Q., Yan, X. B.: The continuity and regularity of three-point transition for two-parameter jump processes. Journal of Changsha University of Electric Power Natural science, (in Chinese), 11(4), 337–341 (1996)
[27] Yan, X. B., Li, Y. Q.: The differentiability of three-point transition function for two-parameter jump processes (I). Journal of Changsha University of Electric Power Natural Science, (in Chinese), 13(3), 225–228 (1998) · Zbl 0934.60065
[28] Yan, X. B., Li, Y. Q.: The differentiability of three-point transition function for two-parameter jump processes(II). Journal of Changsha University of Electric Power Natural Science, (in Chinese), 13(4), 337–342 (1998)
[29] Zhang, J. P.: N-parameter d-dimension random walk(RW n d ). Acta Mathematica Sinica, Chinese Series, 43(3), 517–525 (2000)
[30] Xie, Y. Q.: Standard tri-point transition function. Science in China, Series A., 48(7), 904–914 (2005) · Zbl 1136.60354
[31] Xie, Y. Q.: Tri-point transition function with three states. Journal of Engineering mathematics, (in Chinese), 23(4), 733–736 (2006) · Zbl 1115.60322
[32] Vestrup, E. M.: The theory of measures and integration, John Wiley and Sons, 1971 · Zbl 1059.28001
[33] Brooks, J. K.: On te Vitali-Hahn-Saks and Nikodym theorem. Mathematics Proc., 64, 468–472 (1967)
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