Abstract
We consider a continuous-time branching random walk on the integer lattice ℤd (d ≥ 1 ) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs.
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Albeverio, S., Bogachev, L.V., and Yarovaya, E.B.: Asymptotics of branching symmetric random walk on the lattice with a single source, C. R. Acad. Sci. Paris, Sér. I, Math. 326 (1998), 975–980; Erratum, Ibid. 327 (1998), 585.
Albeverio, S., Bogachev, L.V., and Yarovaya, E.B.: Branching random walk with a single source: A moment approach, Preprint, SFB 256, Universität Bonn, 1998. [To appear in: S. Elaydi, G. Ladas, and J. Popenda (eds), Proceedings of the Fourth International Conference on Difference Equations and Applications, August 27–31, 1998, Pozna´n, Poland, Gordon and Breach Publ.]
Athreya, K.B. and Ney, P.E.: Branching Processes, Springer, Berlin–Heidelberg–New York, 1972.
Dawson, D.A.: Measure-valued Markov processes, in: P.L. Hennequin (ed.), Ecole d'Eté de Probabilités de Saint-Flour XXI-1991, Lect. Notes Math., 1541, Springer, Berlin–Heidelberg–New York, 1993, pp. 1–260.
Daleckii, Yu. L. and Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space, American Math. Soc., Providence, R.I., 1974.
Dawson, D.A., Fleischmann, K., and Le Gall, J.F.: Super-Brownian motions in catalytic media, in: C.C. Heyde (ed.), Branching Processes, Proc. First World Congress, Lect. Notes Stat., 99, Springer, Berlin–Heidelberg–New York, 1995, pp. 122–134.
Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, New York–London–Sydney, 1968.
Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., Wiley, New York–London–Sydney, 1971.
Fleischmann, K.: Superprocesses in catalytic media, in: D.A. Dawson (ed.), Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems, CRM Proceedings and Lecture Notes, 5, American Math. Soc., Providence, R.I., 1994, pp. 163–180.
Gihman, I.I. and Skorohod, A.V.: The Theory of Stochastic Processes, Vol. II, Nauka, Moscow,1973 (in Russian); English translation: The Theory of Stochastic Processes II, Springer, Berlin–Heidelberg–New York, 1975.
Halmos, P.R.: A Hilbert Space Problem Book, Van Nostrand, Princeton, N.J.–Toronto–London, 1967.
Harris, T.E.: The Theory of Branching Processes, Springer, Berlin–Göttingen–Heidelberg,1963.
Hille, E.: Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass.– Menlo Park, Calif.–London–Don Mills, Ontario, 1969.
Jordan, Ch.: Calculus of Finite Differences, 3rd ed., Chelsea Publ. Co., New York, 1965.
Molchanov, S.A.: Lectures on random media, in: D. Bakry, R.D. Gill, and S.A. Molchanov, Lectures on Probability Theory, Ecole d'Eté de Probabilités de Saint-Flour XXII-1992 (P. Bernard, ed.), Lect. Notes Math., 1581, Springer, Berlin–Heidelberg–New York, 1994, pp. 242–411.
Moyal, J.E.: The general theory of stochastic population processes, Acta Math. 108 (1962), 1–31.
Sevastyanov, B.A.: Branching Processes, Nauka, Moscow, 1971 (in Russian); German translation: Sewastjanow, B.A.: Verzweigungsprozesse, Akademie-Verlag, Berlin, 1974, and Oldenbourg Verlag, München–Wien, 1975.
Shubin, M.A.: Pseudo-difference operators and their Green functions, Izv. Akad. Nauk SSSR, Ser. Mat. 49 (1985), 652–671 (in Russian); English translation: Math. USSR, Izv. 26 (1986), 605–622.
Spitzer, F.: Principles of Random Walk, 2nd ed., Springer, New York–Heidelberg–Berlin, 1976.
Vatutin, V.A. and Zubkov, A.M.: Branching processes, I, in: Itogi Nauki Tekhn., Ser. Teor. Veroyatn., Mat. Stat., Teor. Kibern. 23 (1985), 3–67 (in Russian); English translation: J. Sov. Math. 39 (1987), 2431–2475.
Vatutin, V.A. and Zubkov, A.M.: Branching processes, II,J. Sov. Math. 67 (1993), 3407–3485.
Yarovaya, E.B.: Use of spectral methods to study branching processes with diffusion in a noncompact phase space, Teor. Mat. Fiz. 88 (1991), 25–30 (in Russian); English translation: Theor. Math. Phys. 88 (1991), 690–694.
Zhao, X.-L.: On a class of measure-valued processes with non-constant branching rates, in: Z.M. Ma, M. Röckner, and J.A. Yan (eds), Dirichlet Forms and Stochastic Processes, Proc. Internat. Conf., Beijing, China, October 25–31, 1993, Walter de Gruyter, Berlin–New York, 1995, pp. 425–433.
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Albeverio, S., Bogachev, L.V. Branching Random Walk in a Catalytic Medium. I. Basic Equations. Positivity 4, 41–100 (2000). https://doi.org/10.1023/A:1009818620550
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DOI: https://doi.org/10.1023/A:1009818620550