Tightness for branching random walk in time-inhomogeneous random environment

TL;DR

This paper studies a branching random walk in a time-inhomogeneous environment with random branching, proving that the maximum particle position, after suitable centering, remains tightly bounded with high probability.

Contribution

It establishes tightness results for the maximum particle position in a branching random walk with i.i.d. random branching factors and Gaussian increments, in a time-inhomogeneous setting.

Findings

The maximum particle position, after centering, is tight with high probability.

The centering functions depend only on the branching factors up to current generation.

The model extends classical branching random walk results to a random environment setting.

Abstract

We consider a branching random walk in time-inhomogeneous random environment, in which all particles at generation $k$ branch into the same random number of particles $\mathcal{L}_{k+1}\ge 2$, where the $\mathcal{L}_k$, $k\in\mathbb{N}$, are i.i.d., and the increments are standard normal. Let $\mathbb{P}$ denote the law of $(\mathcal{L}_k)_{k\in\mathbb{N}}$, and let $M_n$ denote the position of the maximal particle in generation $n$. We prove that there are $m_n$, which are functions of only $(\mathcal{L}_k)_{k\in\{0,\dots, n\}}$, such that (with regard to $\mathbb{P}$) the sequence $(M_n-m_n)_{n\in\mathbb{N}}$ is tight with high probability.