WIAS Preprint No. 1861, (2013)

Moment asymptotics for multitype branching random walks in random environment



Authors

  • Gün, Onur
  • König, Wolfgang
  • Sekulović, Ozren

2010 Mathematics Subject Classification

  • 60J80 60J55 60F10 60K37 60J10

Keywords

  • multitype branching random walk, Feynman-Kac-type formula, variational analysis, annealed moments, large deviations

Abstract

We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/ρij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Appeared in

  • J. Theoret. Probab., 28 (2015) pp. 1726--1742.

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