Singular SPDEs and fluctuations of particle systems

Abstract

This thesis studies the large scale behaviour of biological processes in a random en- vironment. We start by considering a system of branching random walks in which the branching rates are determined by a random spatial catalyst. In an appropriate setting we show that this process converges to a superBrownian motion in a space white noise potential. We study the asymptotic properties of this superprocess and prove that it survives with positive probability. We then consider scaling limits of a spatial Λ–Fleming–Viot model, relating it both to the process we just introduced and to a stochastic Fisher-KPP equation. Finally, we study the longtime behaviour of the Kardar–Parisi–Zhang equation on finite volume, proving asymptotic synchroniza- tion and a one force, one solution principle. Our analyses rely on techniques from singular stochastic partial differential equations for the parabolic Anderson model and the KPZ equation, and on the theory of superprocesses.