A Rough Superbrownian Motion

We study the large scale behavior of a branching random walk in a random environment (particles move independently and branch at a state-dependent rate provided by a random potential). We introduce a density parameter and as this parameter assumes a critical value, we can prove the scaling limit of the particle system to a rough analogous of the Superbrownian motion. We can characterize this process in dimension d=1 via a nonlinear stochastic PDE. We also study some statistical properties of the process.