A Rough Super-Brownian Motion

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: $${\mathrm{\partial }}_t\mu =(\mathrm{\Delta }+\xi )\mu +\sqrt{2\nu \mu }\tilde{\xi }$$ 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.