On a class of infinite particle systems evolving in a random environment

Abstract

We consider a class of processes on (ℕ)^S (S=ℤⁿ), modelling population growth. The dynamics of the system consists of: motion of particles, birth and death of individual particles, extinction of all particles at a site and splitting of all particles at a site.

We investigate the changes in the longterm behaviour of these systems, changes, which occur if we replace parameters of the evolution (as offspring distribution or site killing rate) by collections indexed by the sites and generated by a random mechanism at time 0.

We study the system for each (a.s.) realisation of the random environment and show that the exponential growth rate of the expected number of particles per site (given the environment) depends heavily on the character of the underlying motion; the growth rate is maximal (and can be calculated explicitly) iff the underlying motion has no drift. We propose an approach for the more detailed study of the asymptotic behaviour (t→∞) of the process and show for Branching Random Walks a law of large numbers, respectively convergence to a "Poisson limit". Furthermore we show that nontrivial equilibria for our evolutions can exist only in the case of a translation-invariant structure of the mean offspring size and the mean death rates.