Electronic Journal of Probability

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

Anja Sturm

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 6, 39 p.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464037579

Digital Object Identifier
doi:10.1214/EJP.v8-129

Mathematical Reviews number (MathSciNet)
MR1986838

Zentralblatt MATH identifier
1064.60199

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems

Keywords
Heat equation colored noise stochastic partial differential equation superprocess weak convergence particle representation random environment existence theorem

Citation

Sturm, Anja. On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise. Electron. J. Probab. 8 (2003), paper no. 6, 39 p. doi:10.1214/EJP.v8-129. https://projecteuclid.org/euclid.ejp/1464037579


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