A Rough Super-Brownian Motion

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: \[\partial_t μ= (Δ{+} ξ) μ{+} \sqrt{2νμ} \tildeξ\] for independent space white noise $\hat{I}¾$ and space-time white noise $\tilde{\hat{I}}¾$. 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.

30 Pages. This is a significantly shortened version of the original, a part of which was migrated to the article named "Killed rough super-Brownian motion"