A distribution-function-valued SPDE and its applications

Abstract

In this paper we further study the stochastic partial differential equation first proposed by Xiong [22]. Under localized conditions on its coefficients, we prove a comparison theorem on its solutions and show that the solution is in fact distribution-function-valued. We also establish pathwise uniqueness of the solution. As applications we obtain the well-posedness of martingale problems for two classes of measure-valued diffusions: interacting super-Brownian motions and interacting Fleming–Viot processes. Properties of the two superprocesses such as the existence of density fields and the survival–extinction behaviors are also studied.