A brief survey of superprocesses over a stochastic flow

In this chapter, we survey the recent progress in the study of superprocesses over a stochastic flow. Firstly, the process is defined as the high-density limit of a branching particle system in random environment. Then, the log-Laplace transform is characterized by a nonlinear SPDE. Thirdly, this measure-valued process has a density when d = 1 which is governed by an SPDE. Making use of Krylov’s Lp theory, the continuity of the density is then established. Fourthly, the longtime behavior of the process is studied. It has persistency for higher spatial dimension and local extinction for lower ones. Finally, the limit behavior of the occupation measure is investigated. An ergodic property is proved for d ≥ 3 and a fluctuation result is derived when d = 2.