Supercritical branching diffusions in random environment

TL;DR

This paper explores the behavior of supercritical branching diffusions in random environments, revealing a phase transition similar to subcritical processes when conditioned on extinction, and clarifies their probabilistic laws.

Contribution

It demonstrates that conditioned supercritical BDREs share the same law as subcritical BDREs, highlighting a phase transition phenomenon in random environments.

Findings

Conditioned supercritical BDREs have the same law as subcritical BDREs.

A phase transition occurs in supercritical BDREs conditioned on extinction.

The behavior of BDREs in random environments differs from constant environments.

Abstract

Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population is not a BDRE. However the quenched law of the population size of a supercritical BDRE conditioned on eventual extinction is equal to the quenched law of the population size of a subcritical BDRE. As a consequence, supercritical BDREs have a phase transition which is similar to a well-known phase transition of subcritical branching processes in random environment.