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2003 On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise
Anja Sturm
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Electron. J. Probab. 8: 1-39 (2003). 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.

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Anja Sturm. "On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise." Electron. J. Probab. 8 1 - 39, 2003. https://doi.org/10.1214/EJP.v8-129

Information

Published: 2003
First available in Project Euclid: 23 May 2016

zbMATH: 1064.60199
MathSciNet: MR1986838
Digital Object Identifier: 10.1214/EJP.v8-129

Subjects:
Primary: 60H15
Secondary: 60F05 , 60J80 , 60K35 , 60K37

Keywords: colored noise , existence theorem , heat equation , particle representation , random environment , Stochastic partial differential equation , Superprocess , weak convergence

Rights: Copyright © 2003 The Institute of Mathematical Statistics and the Bernoulli Society

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