(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Gün, Onur; König, Wolfgang; Sekulovic, Ozren
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/pij 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.
