# Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

- Authors
- Type
- Preprint
- Publication Date
- Dec 22, 2021
- Submission Date
- Jul 31, 2015
- Identifiers
- DOI: 10.1016/j.spa.2018.09.008
- Source
- arXiv
- License
- Yellow
- External links

## Abstract

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures' laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form $V_n -\varphi \log n + o_\mathbf{P}(\log n)$, where $V_n$ is a function of the environment that behaves as a random walk and $\varphi>0$ is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.