Stochastic averaging for a spatial population model in random environment

TL;DR

This paper investigates the non-equilibrium dynamics of a spatial population model with two particle types in a random environment, proving that as the environmental interaction rate increases, the system's evolution converges to an averaged state.

Contribution

It introduces a stochastic averaging method for a complex spatial population model with environment interactions, establishing weak convergence of the state evolution as environmental effects are averaged out.

Findings

Weak convergence of the state evolution as environmental interaction rate tends to zero.

Development of a correlation function evolution framework on Banach spaces.

Identification of the limiting dynamics as an averaged process over the environment.

Abstract

In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations $\Gamma^2 = \Gamma^+ \times \Gamma^-$ 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 $(\mu_t^{\varepsilon})_{t \geq 0}$ is obtained from the Fokker-Planck equation with Markov operator $L^S + \frac{1}{\varepsilon}L^E$, $\varepsilon > 0$, 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 $\varepsilon \to 0$ the state evolution $\mu_t^{\varepsilon}$ converges weakly to some state evolution $\overline{\mu}_t$ 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.