Summary
Markov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero, i.e. the conditional expectation of sojourn time at zero is zero. In this paper we consider the existence and uniqueness of such a structure. We prove that if the sum of the immigration rates is finite then no such structure can exist, and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite. Study of the uniqueness problem shows that for honest processes the solution is unique.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Athreya, K.B., Ney, P.E.: Branching processes. Berlin Heidelberg New York: Springer 1972
Chung, K.L.: Markov chains with stationary transition probabilities. Berlin Heidelberg New York: Springer 1967
Chung, K.L.: Lectures on boundary theory for Markov chains. Ann. Math. Stud.65 (1970)
Doney, R.A.: A note on some results of Schuh. J. Appl. Probab.21, 192–196 (1984)
Doob, J.L.: Markoff chains-denumerable case. Trans. Am. Math. Soc.58, 455–473 (1945)
Doob, J.L.: Stochastic processes. New York: Wiley 1953
Feller, W.: On the integro-differential equations of purely discontinuous Markov processes. Trans. Am. Math. Soc.48, 488–515 (1940)
Foster, J.H.: A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist.42, 1773–1776 (1971)
Freedman, D.: Markov chains. Berlin Heidelberg New York: Springer 1983
Harris, T.E.: The theory of branching processes. Berlin Heidelberg New York: Springer 1963
Hille, E.: Functional analysis and semi-groups. Colloq. Publ., Am. Math. Soc. (1948)
Kemeny, J.G., Snell, J., Knapp, A.W.: Denumerable Markov chains. Princeton: Van Nostrand 1966
Kendall, D.G.: Some analytical properties of continuous stationary Markov transition functions. Trans. Am. Math. Soc.78, 529–540 (1955)
Kendall, D.G., Reuter, G.E.H.: Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators onl. Proc. Intern. Congr. Math. Amsterdam, Vol. III, pp. 377–415 (1954)
Kolmogorov, A.N.: On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov. Gos. Univ. Učenye Zapiski Matematika.148, 53–59 (1951)
Lévy, P.: Complément à l'étude des processus de Markoff. Ann. Sci. Éc. Norm. Supér. (3),69, 203–212 (1952)
Mitov, K.V., Vatutin, V.A., Yanev, N.M.: Continuous-time branching processes with decreasing state-dependent immigration. Adv. Appl. Probab.16, 697–714 (1984)
Neveu, J.: Lattice methods and submarkovian processes. Proc. 4th Berk. Symp. Math. Statist. Prob., Vol 2, pp. 347–391. University of California Press (1960)
Pakes, A.G.: A branching process with a state-dependent immigration component. Adv. Appl. Probab.3, 301–314 (1971)
Pakes, A.G.: Some results for non-supercritical Galton-Watson processes with immigration. Math. Biosci. 24, 71–92 (1975)
Pakes, A.G.: On the age distribution of a Markov chain. J. Appl. Probab.15, 65–77 (1978)
Reuter, G.E.H.: Denumerable Markov processes and the associated semigroup onl. Acta. Math.97, 1–46 (1957)
Reuter, G.E.H.: Denumerable Markov processes (II). J. Lond. Math. Soc.34, 81–91 (1959)
Reuter, G.E.H.: Denumerable Markov processes (III). J. Lond. Math. Soc.37, 63–73 (1962)
Reuter, G.E.H.: Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitstheor. Verw. Geb.13, 315–320 (1969)
Schuh, H.J.: Sums of i.i.d. random variables and an application to the explosion criterion for Markov branching processes. J. Appl. Probab.19, 29–38 (1982)
Seneta, E.: Regularly varying functions. (Lect. Notes Math. vol. 508) Berlin Heidelberg New York: Springer 1976
Williams, D.: A note on theQ-matrices of Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb.7, 116–121 (1967)
Williams, D.: Diffusions, Markov processes and martingales, vol. 1. Foundations. New York: Wiley 1979
Yamazato, M.: Some results on continuous time branching processes with state-dependent immigration. J. Math. Soc. Japan,27, 479–496 (1975)
Yosida, K.: On the differentiability and the representation of one-parameter semi-groups of linear operators. J. Math. Soc. Japan,1, 15–21 (1948)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, A.Y., Renshaw, E. Markov branching processes with instantaneous immigration. Probab. Th. Rel. Fields 87, 209–240 (1990). https://doi.org/10.1007/BF01198430
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01198430