A rough super-Brownian motion

Abstract

We study the scaling limit of a branching random walk in static random environment in dimension $\mathit{d}=1,2$ 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

$${\partial }_{\mathit{t}}\mathit{\mu }=(\mathrm{\Delta }+\mathit{\xi })\mathit{\mu }+\sqrt{2\mathit{\nu }\mathit{\mu }}\tilde{\mathit{\xi }}$$

for independent space white noise ξ and space-time white noise $\tilde{\mathit{\xi }}$. 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.