Electronic Journal of Probability
- Electron. J. Probab.
- Volume 8 (2003), paper no. 6, 39 p.
On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise
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.
Electron. J. Probab., Volume 8 (2003), paper no. 6, 39 p.
First available in Project Euclid: 23 May 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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