On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

Anja Sturm

doi:10.1214/ejp.v8-129

Abstract

We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

Highlights

The stochastic heat equation considered in this paper is a stochastic partial differential equation (SPDE), which can formally be written as
U is a random function on R+ × Rd, where R+ ≡ [0, ∞), and the operator ∆ denotes the Laplacian acting on Rd
We are concerned with convergence of rescaled branching particle systems in a random environment and associated lattice systems, which are infinite systems of stochastic differential equations (SDE), to solutions of (1)

Summary

Introduction

The stochastic heat equation considered in this paper is a stochastic partial differential equation (SPDE), which can formally be written as. (i) The heat equation with a multiplicative noise term that is white in space and time arises in studying the diffusion limit of a large class of spatially distributed (for the most part branching) particle systems. (ii) Stochastic heat equations of the form (1), where W is white in space and time, have function valued solutions only in dimension one Connections of these SPDEs to population systems are restricted to a one dimensional state space. The approximation by a system of SDEs leads to a new existence result for the stochastic heat equation with colored noise and non-Lipschitz noise coefficients Both representations may be exploited further for numerical purposes or the study of properties of these SPDEs. In the following we elucidate these points a bit further and point out connections to related work.

Formulation of the main results

Construction of the particle system

Convergence to the heat equation

Auxiliary lemmas

Remarks

Continuity of solutions