Branching Processes in Random Environment — A View on Critical and Subcritical Cases

Abstract

Branching processes exhibit a particularly rich longtime behaviour when evolving in a random environment. Then the transition from subcriticality to supercriticality proceeds in several steps, and there occurs a second ‘transition’ in the subcritical phase (besides the phase-transition from (sub)criticality to supercriticality). Here we present and discuss limit laws for branching processes in critical and subcritical i.i.d. environment. The results rely on a stimulating interplay between branching process theory and random walk theory. We also consider a spatial version of branching processes in random environment for which we derive extinction and ultimate survival criteria.