Fleming-Viot Processes in an Environment

TL;DR

This paper introduces a new class of measure-valued processes influenced by individual and common environmental noise, deriving their properties and connecting them to branching diffusions and superprocesses.

Contribution

It constructs a novel type of measure-valued process influenced by environment-like noise and derives associated stochastic PDEs, extending classical Fleming-Viot process theory.

Findings

Process values are either purely atomic or absolutely continuous.

Derived a new stochastic PDE for the density when absolutely continuous.

Connected these processes to measure-valued branching diffusions in a Brownian medium.

Abstract

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line $\mbb{R}$ are constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson-Watanabe superprocesses, conditioned to have total mass one, are Fleming-Viot superprocesses.