Branching random walks in random environment and super-Brownian motion in random environment

TL;DR

This paper studies the limiting behavior of critical branching random walks in a random environment, showing they converge to a super-Brownian motion in a random environment, and establishes uniqueness of solutions.

Contribution

It characterizes the weak limit of branching random walks in random environment as a super-Brownian motion and proves weak uniqueness of solutions.

Findings

Weak convergence to super-Brownian motion in random environment

Solution characterized as a martingale problem

Weak uniqueness of solutions under certain initial conditions

Abstract

We focus on the existence and characterization of the limit for a certain critical branching random walks in time-space random environment in one dimension which was introduced by M. Birnkenr et.al. Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time-space site. The weak limit of this measure valued processes is characterized as a solution to the non-trivial martingale problem and called super-Brownian motions in random environment by L. Mytnik. Moreover, we will show the weak uniqueness of the solutions with some initial condition.