Moment asymptotics for multitype branching random walks in random environment
- Gün, Onur
- König, Wolfgang
- Sekulović, Ozren
2010 Mathematics Subject Classification
- 60J80 60J55 60F10 60K37 60J10
- multitype branching random walk, Feynman-Kac-type formula, variational analysis, annealed moments, large deviations
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.
- J. Theoret. Probab., 28 (2015) pp. 1726--1742.