# A Rough Super-Brownian Motion

## TL;DR

This paper investigates the scaling limit of a branching random walk in static random environments in one and two dimensions, showing it converges to a super-Brownian motion influenced by white noise, with detailed characterizations in each dimension.

## Contribution

It introduces a novel rough super-Brownian motion as the scaling limit of branching random walks in random environments, using stochastic PDEs and martingale problems for characterization.

## Key findings

- In dimension 1, the limit is characterized by a stochastic PDE with white noise.
- In dimension 2, the limit process is described via a martingale problem.
- Persistence of the rough super-Brownian motion is established in both dimensions.

## Abstract

We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the unique weak solution to the stochastic PDE: \[\partial_t \mu = (\Delta {+} \xi) \mu {+} \sqrt{2\nu \mu} \tilde{\xi}\] for independent space white noise $\xi$ and space-time white noise $\tilde{\xi}$. In dimension $2$ the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-Brownian motion.