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Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. (English) Zbl 0791.60063
Until now, denumerable Markov processes with instantaneous states have not been extensively considered, and so we present a detailed examination of the conservative uni-instantaneous (CUI) case. We determine criteria for the existence and uniqueness of a specific CUI pregenerator, and consider the general problem of constructing CUI processes.
Reviewer: A.Chen

MSC:
60J27Continuous-time Markov processes on discrete state spaces
References:
[1]Blackwell, D.: Another countable Markov process with only instantaneous states. Ann. Math. Stat.29, 313-316 (1958) · Zbl 0085.12702 · doi:10.1214/aoms/1177706735
[2]Chen, A.Y., Renshaw, E.: Markov branching processes with instantaneous immigration. Probab. Theory Relat. Fields87, 209-240 (1990) · Zbl 0695.60080 · doi:10.1007/BF01198430
[3]Chung, K.L.: On the boundary theory for Markov chains. Acta. Math.115, 111-163 (1966). · Zbl 0315.60041 · doi:10.1007/BF02392205
[4]Chung, K.L.: Markov chains with stationary transition probabilities. Berlin Heidelberg New York: Springer 1967
[5]Chung, K.L.: Lectures on boundary theory for Markov chains. (Ann. Math. Stud., 65) Princeton University Press 1970
[6]Cinlar, E.: Introduction to stochastic processes. Englewood Cliffs, NY: Prentice-Hall 1975
[7]Doob, J.L.: Markoff chains-Denumerable case. Trans. Am. Math. Soc.58, 455-473 (1945)
[8]Feller, W.: On the integro-differential equations of purely discontinuous Markov processes. Trans. Am. Math. Soc.48, 488-515 (1940) · doi:10.1090/S0002-9947-1940-0002697-3
[9]Feller, W.: On boundaries and lateral conditions for the Kolmogoroff differential equations. Ann. Math.65, 527-570 (1957) · Zbl 0084.35503 · doi:10.2307/1970064
[10]Feller, W.: The birth and death processes as diffusion processes. J. Math. Pures Appl.9, 301-345 (1959)
[11]Freedman, D.: Markov chains. Berlin Heidelberg New York: Springer 1983
[12]Freedman, D.: Approximating countable Markov chains. Berlin Heidelberg New York: Springer 1983
[13]Hou, Z.T.: The uniqueness criterion for Q-processes (in Chinese). Sinica2, 115-130 (1974)
[14]Hou, Z.T., Guo, Q.F.: Homogeneous denumerable Markov processes. Berlin Heidelberg New York: Springer 1988
[15]Kemeny, J.G., Snell, J., Knapp, A.W.: Denumerable Markov chains. Princeton, NJ: Van Nostrand 1966
[16]Kendall, D.G.: Some analytical properties of continuous stationary Markov transition functions. Trans. Am. Math. Soc78, 529-540 (1955) · doi:10.1090/S0002-9947-1955-0067401-2
[17]Kendall, D.G.: A totally unstable denumerable Markov process. Q. J. Math., Oxford9, 149-160 (1958) · Zbl 0081.13301 · doi:10.1093/qmath/9.1.149
[18]Kendall, D.G., Reuter, G.E.H.: Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators on ?. Proc. Intern. Congr. Math. Amsterdam, Vol. III, 377-415. Amsterdam: North-Holland 1954
[19]Kolmogorov, A.N.: On the differentiability of the transition probabilities in homogeneous Markov processes with a denumerable number of states. Moskov. Gos. Univ. U?enye Zapiski MGY. 148 Mat.4, 53-59 (1951)
[20]Neveu, J.: Lattice methods and submarkovian processes. Proc. 4th Berk. In: Symp. Math. Statist. Prob., vol 2, pp. 347-391. Berkeley, CA: University of California Press 1960
[21]Reuter, G.E.H.: Denumerable Markov processes and the associated semigroup on ?. Acta. Math.97, 1-46 (1957) · Zbl 0079.34703 · doi:10.1007/BF02392391
[22]Reuter, G.E.H.: Denumerable Markov processes (II). J. Lond. Math. Soc.34, 81-91 (1959) · Zbl 0089.13803 · doi:10.1112/jlms/s1-34.1.81
[23]Reuter, G.E.H.: Denumerable Markov processes (III). J. Lond. Math. Soc.37, 63-73 (1962) · Zbl 0114.33604 · doi:10.1112/jlms/s1-37.1.63
[24]Reuter, G.E.H.: Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitstheor. Verw. Geb.13, 315-320 (1969) · Zbl 0176.47803 · doi:10.1007/BF00539207
[25]Reuter, G.E.H.: Denumerable Markov processes (IV). On C.T. Hou’s uniqueness theorem for Q-semigroups. Z. Wahrscheinlichkeitstheor. Verw. Geb.33, 309-315 (1976) · Zbl 0361.60041 · doi:10.1007/BF00534781
[26]Rogers, L.C.G., Williams, D.: Construction and approximation of transition matrix functions. Adv. Appl. Probab., Suppl. 133-160 (1986)
[27]Tang, L.Q.: The construction of uni-instantaneous birth and death processes. Chin. Ann. Math.8A(5), 565-570 (1987)
[28]Williams, D.: On the construction problem for Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 227-246 (1964) · Zbl 0143.19804 · doi:10.1007/BF00534911
[29]Williams, D.: A note on the Q-matrices of Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb.7, 116-121 (1967) · Zbl 0178.20304 · doi:10.1007/BF00536325
[30]Williams, D.: The Q-matrix problem. In: Meye, P.A. (ed.) Séminaire de Probabilities X: (Lect. Notes Math., vol 511, pp. 216-234) Berlin Heidelberg New York: Springer 1976
[31]Williams, D.: Diffusions, Markov processes and martingales, vol 1, Foundations. New York: Wiley 1979
[32]Yang, X.Q.: The construction theory of denumerable Markov processes. New York: Wiley 1990