Co-existence of branching populations in random environment
Nikita Elizarov, Vitali Wachtel
TL;DR
The paper addresses the coexistence of two branching processes in a joint random environment and proves that the probability both populations survive up to time $n$ decays polynomially with exponent $\theta(\varrho)=\frac{\pi}{2\arccos(-\varrho)}$, reflecting environmental dependence through the covariance $\varrho$. The authors connect this probability to the behavior of an associated two-dimensional random walk confined to the positive quadrant, employ a linear transformation to obtain uncorrelated coordinates, and use a harmonic function $V$ together with Doob's $h$-transform to establish entropic repulsion and the asymptotics. A key contribution is the explicit exponent formula, positivity of the leading constant, and a conditional limit theorem describing the growth path of coexisting populations in terms of a Brownian meander in $\mathbb{R}_+^2$. The results advance understanding of multi-type branching processes in random environments and provide techniques relevant to stochastic ecological models under environmental randomness.
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 $n^{-θ}$ with $θ>0$ 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.