Co-existence of branching populations in random environment

TL;DR

This paper investigates the survival probability of two critical branching processes in a shared random environment, revealing it decays polynomially with time and establishing a related limit theorem.

Contribution

It introduces a novel analysis of joint survival probabilities in critical branching processes within a random environment, including a new bound on entropic repulsion for 2D random walks.

Findings

Survival probability decays as n^(-θ) with θ>0 depending on the environment

Established a conditional limit theorem for the processes

Provided a new bound on entropic repulsion in positive quadrant

Abstract

In this paper we consider two branching processes living in a joint random environment. Assuming that both processes are critical we address the following question: What is the probability that both populations survive up to a large time $n$? We show that this probability decays as $n^{-\theta}$ with $\theta>0$ which is determined by the random environment. Furthermore, we prove the corresponding conditional limit theorem. One of the main ingredients in the proof is a qualitative bound for the entropic repulsion for two-dimensional random walks conditioned to stay in the positive quadrant. We believe that this bound is also of independent interest.