## On multidimensional branching random walks in random environment.(English)Zbl 1114.60084

Summary: We study branching random walks in random i.i.d. environment in $$\mathbb{Z}^d$$, $$d\geq 1$$. For this model, the population size cannot decrease, and a natural ural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.

### MSC:

 60K37 Processes in random environments 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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### References:

 [1] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546. · Zbl 1013.60081 [2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002 [3] Baillon, J.-B., Clément, P., Greven, A. and den Hollander, F. (1993). A variational approach to branching random walk in random environment. Ann. Probab . 21 290–317. · Zbl 0770.60088 [4] Biggins, J. (1978). The asymptotic shape of the branching random walk. Adv. in Appl. Probab . 10 62–84. JSTOR: · Zbl 0383.60078 [5] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumor growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357. · Zbl 0459.92013 [6] Comets, F., Menshikov, M. V. and Popov, S. Yu. (1998). One-dimensional branching random walk in random environment: A classification. Markov Process. Related Fields 4 465–477. · Zbl 0938.60081 [7] Dembo, A., Peres, Y. and Zeitouni, O. (1996). Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667–683. · Zbl 0868.60058 [8] Devulder, A. (2005). A branching system of random walks in random environment. Available at http://www.proba.jussieu.fr/mathdoc/textes/PMA-834.pdf. · Zbl 1138.60341 [9] Durrett, R. and Griffeath, D. (1982). Contact processes in several dimensions. Z. Wahrsch. Verw. Gebiete 59 535–552. · Zbl 0483.60089 [10] Engländer, J. (2005). Branching Brownian motion with ‘mild’ Poissonian obstacles. Available at http://arxiv.org/math.PR/0508585. [11] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press. · Zbl 0823.60053 [12] Gantert, N. and Müller, S. (2005). The critical branching random walk is transient. Available at http://arxiv.org/math.PR/0510556. · Zbl 1115.60077 [13] Gantert, N. and Zeitouni, O. (1998). Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177–190. · Zbl 0982.60037 [14] Greven, A. and den Hollander, F. (1992). Branching random walk in random environment: Phase transitions for local and global growth rates. Probab. Theory Related Fields 91 195–249. · Zbl 0744.60079 [15] den Hollander, F., Menshikov, M. V. and Popov, S. Yu. (1999). A note on transience versus recurrence for a branching random walk in random environment. J. Statist. Phys. 95 587–614. · Zbl 0933.60089 [16] Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883–909. JSTOR: · Zbl 0311.60018 [17] Lawler, G. F. (1983). A discrete stochastic integral inequality and balanced random walk in a random environment. Duke Math. J. 50 1261–1274. · Zbl 0569.60071 [18] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Probab. 13 1279–1285. · Zbl 0579.60023 [19] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078 [20] Machado, F. P. and Popov, S. Yu. (2000). One-dimensional branching random walk in a Markovian random environment. J. Appl. Probab. 37 1157–1163. · Zbl 0995.60070 [21] Machado, F. P. and Popov, S. Yu. (2003). Branching random walk in random environment on trees. Stochastic Process. Appl. 106 95–106. · Zbl 1075.60570 [22] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745–789. · Zbl 0418.60033 [23] Pisztora, A. and Povel, T. (1999). Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27 1389–1413. · Zbl 0964.60056 [24] Pisztora, A., Povel, T. and Zeitouni, O. (1999). Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields 113 191–219. · Zbl 0922.60059 [25] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287–323. · Zbl 0947.60095 [26] Sznitman, A.-S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 93–143. · Zbl 0976.60097 [27] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544. · Zbl 0995.60097 [28] Sznitman, A.-S. (2003). On new examples of ballistic random walks in random environment. Ann. Probab. 31 285–322. · Zbl 1017.60104 [29] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869. · Zbl 0965.60100 [30] Varadhan, S. R. S. (2003). Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 1222–1245. · Zbl 1042.60071 [31] Volkov, S. (2001). Branching random walk in random environment: Fully quenched case. Markov Process. Related Fields 7 349–353. · Zbl 0991.60073 [32] Zeitouni, O. (2004). Random walks in random environment. Lecture Notes in Math. 1837 190–312. Springer, Berlin. · Zbl 1060.60103 [33] Zerner, M. (1998). Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 1446–1476. · Zbl 0937.60095 [34] Zerner, M. (2002). A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Comm. Probab. 7 191–197. · Zbl 1008.60107
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