On convergence of population processes in random environments to the stochastic heat equation with colored noise.

*(English)*Zbl 1064.60199Summary: We consider the stochastic heat equation with a multiplicative colored noise term on \(\mathbb{R}^d\) for \(d\geq 1\). 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.

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60F05 | Central limit and other weak theorems |

60K37 | Processes in random environments |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |