Moment asymptotics for multitype branching random walks in random\n environment

2013/10/29 by Onur Gün, Wolfgang König, Gün, Onur +3
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60F10 #60J10 #60J55 #60J80 #60K37 #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1310.7770

openalex publication_date 2013/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a discrete time multitype branching random walk on a finite space\nwith finite set of types. Particles follow a Markov chain on the spatial space\nwhereas offspring distributions are given by a random field that is fixed\nthroughout the evolution of the particles. Our main interest lies in the\naveraged (annealed) expectation of the population size, and its long-time\nasymptotics. We first derive, for fixed time, a formula for the expected\npopulation size with fixed offspring distributions, which is reminiscent of a\nFeynman-Kac formula. We choose Weibull-type distributions with parameter\n1/\ρ_ij for the upper tail of the mean number of j type particles\nproduced by an i type particle. We derive the first two terms of the\nlong-time asymptotics, which are written as two coupled variational formulas,\nand interpret them in terms of the typical behavior of the system.\n

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