A distribution-function-valued SPDE and its applications

TL;DR

This paper investigates a distribution-function-valued stochastic PDE, proving solution uniqueness and applying results to measure-valued diffusions like super-Brownian motions and Fleming-Viot processes, including their densities and extinction properties.

Contribution

It establishes the well-posedness and pathwise uniqueness of a distribution-function-valued SPDE and applies these results to measure-valued diffusions, expanding understanding of their properties.

Findings

Proves solution is distribution-function-valued under localized conditions

Establishes pathwise uniqueness of the solution

Analyzes properties like density existence and extinction behaviors of superprocesses

Abstract

In this paper we further study the stochastic partial differential equation first proposed by Xiong (2013). Under localized conditions on the coefficients we show that the solution is in fact distribution-function-valued and we establish the pathwise uniqueness of the solution. As applications we obtain the well-posedness of the 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 extinction behaviors are also studied.