Random walks and branching processes in correlated Gaussian environment

TL;DR

This paper investigates the behavior of random walks in correlated Gaussian environments and derives implications for critical branching processes with correlated parameters, focusing on tail distributions and extinction times.

Contribution

It introduces new estimates for persistence probabilities in correlated Gaussian environments and connects these to properties of related branching processes.

Findings

Derived tail distribution estimates for total population size

Analyzed maximum population and extinction time in branching processes

Linked persistence probabilities to properties of correlated Gaussian environments

Abstract

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.