随机环境中空间种群模型的随机平均
Stochastic averaging for a spatial population model in random environment
摘要：
本文研究了空间布居模型在$\mathbb{R}^d$上局部有限组态空间$Γ^2 = Γ^+ \times Γ^-$上的非平衡马尔可夫态演化，其中粒子的自旋为$\pm$ .“+”类型的粒子彼此独立地复制自己，而且由于相同类型的粒子或不同类型的粒子之间的竞争而死亡。粒子类型的演变根据非平衡Glauber型动力学与活动$z$和潜在的$ψ$。设$L^S$为“+"粒子的马尔可夫算子，$L^E$为”-“粒子的马尔可夫算子。非平衡态演化$(μ_t^{\varepsilon})_{t \geq 0}$是从具有马尔可夫算子$L^S + \frac{1}{\varepsilon}L^E$的Fokker-Planck方程得到的，$\varepsilon > 0$本身是根据在适当选择的Banach空间尺度上的相关函数演化来研究的。我们证明了在限制制度$\varepsilon \to 0$的状态演化$μ_t^{\varepsilon}$弱收敛到一些状态演化$\overlineμ_t$相关的福克-普朗克方程与（启发式）马尔可夫算子从$L^S$通过平均系统与环境的相互作用相对于环境的唯一不变的吉布斯测度。
英文摘要：
In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations $Γ^2 = Γ^+ \times Γ^-$ 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 $ψ$. Let $L^S$ be the Markov operator for '+' -particles and $L^E$ the Markov operator for '-' -particles. The non-equilibrium state evolution $(μ_t^{\varepsilon})_{t \geq 0}$ 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 $μ_t^{\varepsilon}$ converges weakly to some state evolution $\overlineμ_t$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
