Stochastic averaging for a spatial population model in random environment

2017/12/09 by Friesen, Martin, Kondratiev, Yuri
#35Q84 #60J80 #60K35 #60K37 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1712.03413

Abstract

In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations Γ² = Γ⁺ × Γ^- over ℝ^d where particles are marked by spins ±. 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 ψ. Let L^S be the Markov operator for '+' -particles and L^E the Markov operator for '-' -particles. The non-equilibrium state evolution (μ_t^ε)_{t ≥ 0} is obtained from the Fokker-Planck equation with Markov operator L^S + (1)/(ε)L^E, ε > 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 ε → 0 the state evolution μ_t^ε converges weakly to some state evolution μ_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.

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