On Convergence of Population Processes in Random Environments
Anja Sturm, Technische Universität Berlin
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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