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 ∂tμ=(Δ+ξ)μ+ 2νμξ˜ for independent space white noise ξ and space-time white noise ξ˜. 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.