Galton-Watson processes in dynamical environments

TL;DR

This paper studies Galton-Watson branching processes in environments that evolve dynamically, analyzing extinction probabilities and environmental regularity, with a focus on critical and supercritical regimes and their mathematical properties.

Contribution

It introduces a model of Galton-Watson processes in dynamical environments and analyzes extinction probabilities and environmental regularity, including Hausdorff dimension of critical sets.

Findings

Extinction probability q in supercritical environments analyzed

Regularity of extinction probability as a function of environment studied

Hausdorff dimension of bad environment set determined in some cases

Abstract

We define a model of Galton Watson processes in dynamical environments where the environment evolves according to a dynamical system (X, T). Three behaviours are possible: uniformly subcritical, critical, and uniformly supercritical. We study the extinction probability q in the uniformly supercritical case. In particular, we investigate the regularity of this application as a function of the environment x. In the critical case, we study the set of bad environments N (where the probability of extinction is one), which is T -invariant. We give its Hausdorff dimension in some cases.