Survival time of random walk in random environment among soft obstacles

Nina Gantert (University of Munster)
Serguei Popov (Universidade de São Paulo)
Marina Vachkovskaia (Universidade de Campinas)


We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general d-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the "mixed" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).

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Pages: 569-593

Publication Date: January 20, 2009

DOI: 10.1214/EJP.v14-631


  1. P. Antal. Enlargement of obstacles for the simple random walk. Ann. Probab. 23 (1995), 1061-1101. Math. Review 96m:60158
  2. G. Ben Arous, S. Molchanov, A.F. Ramírez. Transition from the annealed to the quenched asymptotics for a random walk on random obstacles. Ann. Probab. 33 (2005), 2149-2187. Math. Review 2006k:82067
  3. F. Comets, S. Popov. Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126 (2003), 571-609. Math. Review 2005d:60157
  4. F. Comets, S. Popov. On multidimensional branching random walks in random environment. Ann. Probab. 35 (2007), 68-114. Math. Review 2007k:60336
  5. F. Comets, S. Popov. Shape and local growth for multidimensional branching random walks in random environment. Alea 3 (2007), 273-299. Math. Review 2365644
  6. A. Fribergh, N. Gantert, S. Popov. On slowdown and speedup of transient random walks in random environment. To appear in: Probab. Theory Relat. Fields, arXiv:0806.0790
  7. Y. Hu, Z. Shi. The limits of Sinai's simple random walk in random environment. Ann. Probab. 26 (1998), 1477-1521. Math. Review 2000a:60191
  8. A.S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, (1998). Math. Review 2001h:60147

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