Super-Brownian motion in random environment as a limit point of critical branching random walks in random environment

TL;DR

This paper investigates the limiting behavior of critical branching random walks in a random environment, showing that their weak limits are characterized as super-Brownian motions in random environments.

Contribution

It provides a rigorous characterization of the weak limit points of critical branching random walks in random environments as super-Brownian motions in random environments.

Findings

Weak limit points are solutions to a non-trivial martingale problem.

Limit processes are characterized as super-Brownian motions in random environment.

The study extends understanding of critical branching processes in random media.

Abstract

We focus on the existence and its characterization of limit for a certain critical branching random walks in time-space random environment in 1 dimension which was introduced by Birkner et.al. Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time-space site. The weak limit points of this measure valued processes are characterized as a solution of the non-trivial martingale problem and called super-Brownian motions in random environment by Mytnik.