The Annals of Probability

On multidimensional branching random walks in random environment

Francis Comets and Serguei Popov

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Abstract

We study branching random walks in random i.i.d. environment in ℤd, d≥1. For this model, the population size cannot decrease, and a natural 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.

Article information

Source
Ann. Probab., Volume 35, Number 1 (2007), 68-114.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1174324124

Digital Object Identifier
doi:10.1214/009117906000000926

Mathematical Reviews number (MathSciNet)
MR2303944

Zentralblatt MATH identifier
1114.60084

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Shape theorem recurrence transience subadditive ergodic theorem nestling hitting time

Citation

Comets, Francis; Popov, Serguei. On multidimensional branching random walks in random environment. Ann. Probab. 35 (2007), no. 1, 68--114. doi:10.1214/009117906000000926. https://projecteuclid.org/euclid.aop/1174324124


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