Super-Brownian Motion with Irregular Drift

Abstract: Super-Brownian motion is a stochastic model for the evolution of spatially distributed populations. It arises as a high-density limit of branching Brownian motion and, in one spatial dimension, its density can be described by a stochastic partial differential equation. In this talk, we modify this equation by adding a drift term that models immigration depending on the current local population density. We allow this drift to be irregular, and even discontinuous at zero, which leads to difficulties in proving existence and uniqueness of solutions. We address these questions via the duality method, which relies on the construction of a suitable dual process. In the final part of the talk, we will discuss how the drift affects the long-term behavior of the system, in particular its survival and extinction probabilities.