Tightness for branching random walk in time-inhomogeneous random environment

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.

58 pages, 5 figures, fixed typos, slightly more general formulation of the results in Section 8 for use in future work