Stochastic averaging for a spatial population model in random environment

In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations ${\mathrm{\Gamma }}^2={\mathrm{\Gamma }}^{+}\times {\mathrm{\Gamma }}^{-}$ over ${\mathbb{R}}^d$ where particles are marked by spins $\pm$. Particles of type ‘+’ reproduce themselves independently of each other and, moreover, die due to competition either among particles of the same type or particles of different type. Particles of type ‘-’ evolve according to a non-equilibrium Glauber-type dynamics with activity $z$ and potential $\psi$. Let $L^S$ be the Markov operator for ‘+’ -particles and $L^E$ the Markov operator for ‘-’ -particles. The non-equilibrium state evolution $({\mu }_t^{\epsilon }{)}_{t\ge 0}$ is obtained from the Fokker-Planck equation with Markov operator $L^S+\frac{1}{\epsilon }{L}^E$, $\epsilon >0$, which itself is studied in terms of correlation function evolution on a suitable chosen scale of Banach spaces. We prove that in the limiting regime $\epsilon \to 0$ the state evolution ${\mu }_t^{\epsilon }$ converges weakly to some state evolution ${\bar{\mu }}_t$ associated to the Fokker-Planck equation with (heuristic) Markov operator obtained from $L^S$ by averaging the interactions of the system with the environment with respect to the unique invariant Gibbs measure of the environment.