Abstract
A continuous time branching random walk on the lattice Z is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we study the asymptotic behavior of the joint distribution of the number of individuals at the origin and outside the origin at moment t as t oo given that there are individuals at the origin at this moment1.
Article Footnote
The first author is supported in part by the grants RFBR 03-01-00045, Russian Scientific School 2139.2003.1; the second author is supported in part by the grants RFBR 02-01-00266, Russian Scientific School 1758.2003.1, and INTAS 03-51-5018.
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References
V. A. Topchii, V. A. Vatutin, E. B. Yarovaya, Catalytic branching random walk and queueing systems with random number of independent servers. Theory of Probability and Mathematical Statistics, v. 69, 158–172 (2003).
V. A. Topchii, V. A. Vatutin, Individuals at the origin in the critical catalytic branching random walk. Discrete Mathematics & Theoretical Computer Science, v. 6, 325–332 (2003).
S. Albeverio, L. V. BogachevBranching random walk in a catalytic medium. I. Basic equations. Positivity, v. 4, 41–100 (2000).
S. Albeverio, L. V. Bogachev, E. B. Yarovaya, Asymptotics of branching symmetric random walk on the lattice with a single source. C.-R.-Acad.-Sci.-Paris-Ser.-I-Math., v. 326, 975–980 (1998).
L. V. Bogachev, E. B. Yarovaya, A limit theorem for a supercritical branching random walk on Z d with a single source. Russian-Math.-Surveys, v. 53, 1086–1088 (1998).
L. V. Bogachev, E. B. Yarovaya, Moment analysis of a branching random walk on a lattice with a single source. Dokl.-Akad.-Nauk, v. 363, 439–442 (1998). (In Russian.)
T. E. Harris, (1963) The Theory of Branching Processes Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J.
B. A. Sewastjanow, (1974) Verzweigungsprozesse. Akademie-Verlag, Berlin.
V. A. Vatutin, Critical Bellman-Harris branching processes starting with a large number of particles. Mat.-Zametki, v. 40, 527–541 (1986). (In Russian.)
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Topchii, V., Vatutin, V. (2004). Two-Dimensional Limit Theorem for a Critical Catalytic Branching Random Walk. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds) Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7915-6_39
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DOI: https://doi.org/10.1007/978-3-0348-7915-6_39
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9620-7
Online ISBN: 978-3-0348-7915-6
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