Abstract
We consider a branching random walk in random environment on ℤd where particles perform independent simple random walks and branch, according to a given offspring distribution, at a random subset of sites whose density tends to zero at infinity. Given that initially one particle starts at the origin, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the origin with probability <1 resp. =1. We show that for d≥3 there is a dichotomy in the critical rate of decay, depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and various co-authors investigating the problem of transience versus recurrence for random motions in random media.
Similar content being viewed by others
REFERENCES
F. Comets, M. V. Menshikov, and S. Yu. Popov, Lyapunov functions for random walks and strings in random environment, Ann. Prob. 26:1433–1445.
F. Comets, M. V. Menshikov, and S. Yu. Popov, One-dimensional branching random walk in random environment: A classification, Markov Processes Relat. Fields 4:465–477 (1998).
F. den Hollander, M. V. Menshikov, and S. E. Volkov, Two problems about random walks in a random field of traps, Markov Processes Relat. Fields 1:185–202 (1995).
A. Klenke, A review on spatial catalytic branching, in: International Conference on Stochastic Models, Carleton University, June 9–12, 1998, in honor of D. Dawson, L. Gorostiza and G. Ivanoff, eds. (Conference Proceedings Series of the Canadian Mathematical Society, 1999).
M. V. Menshikov and S. E. Volkov, Branching Markov chains: qualitative characteristics, Markov Processes Relat. Fields 3:1–18 (1997).
R. Pemantle and S. Volkov, Markov chains in a field of traps, J. Theor. Prob. 11:561–569 (1998).
A. Shiryaev, Probability, 2nd ed. (Springer, New York, 1989).
F. Spitzer, Principles of Random Walk, 2nd ed. (Springer, New York, 1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
den Hollander, F., Menshikov, M.V. & Popov, S.Y. A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment. Journal of Statistical Physics 95, 587–614 (1999). https://doi.org/10.1023/A:1004539225064
Issue Date:
DOI: https://doi.org/10.1023/A:1004539225064