连续空间马尔可夫演化的随机平均原理
Stochastic averaging principle for spatial Markov evolutions in the continuum
摘要：
研究了$\mathbb{R}^d$上局部有限位形$Γ^+ \times Γ^-$的相空间上的空间生灭过程.动力学由从Fokker-Planck方程获得的状态的非平衡演化来描述，并与马尔可夫算子$L^+(γ^-) + \frac{1}{\varepsilon}L^-$、$\varepsilon > 0$相关联。这里$L^-$描述$Γ^-$上的环境过程，$L^+(γ^-)$描述$Γ^+$上的系统过程，其中$γ^-$表示相应的生灭率取决于另一个局部有限配置$γ^- \in Γ^-$。我们证明了，对于某类生灭率，相应的Fokker-Planck方程是适定的，即在$Γ^+ \times Γ^-$上存在唯一的态$μ_t^{\varepsilon}$演化.此外，我们给出了一个充分条件，使环境是遍历指数率。设$μ_{\mathrm{inv}}$为$Γ^-$上环境进程的不变测度。在本文的主要部分中，我们建立了随机平均原理，即我们证明了$μ_t^{\varepsilon}$到$Γ^+$上的边际弱收敛于与平均Markov生灭算子$\overline{L} = \int_{Γ^-}L^+(γ^-)d μ_{\mathrm{inv}}(γ^-)$相关的$Γ^+$上的状态演化。
英文摘要：
We study a spatial birth-and-death process on the phase space of locally finite configurations $Γ^+ \times Γ^-$ over $\mathbb{R}^d$. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator $L^+(γ^-) + \frac{1}{\varepsilon}L^-$, $\varepsilon > 0$. Here $L^-$ describes the environment process on $Γ^-$ and $L^+(γ^-)$ describes the system process on $Γ^+$, where $γ^-$ indicates that the corresponding birth-and-death rates depend on another locally finite configuration $γ^- \in Γ^-$. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states $μ_t^{\varepsilon}$ on $Γ^+ \times Γ^-$. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let $μ_{\mathrm{inv}}$ be the invariant measure for the environment process on $Γ^-$. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of $μ_t^{\varepsilon}$ onto $Γ^+$ converges weakly to an evolution of states on $Γ^+$ associated with the averaged Markov birth-and-death operator $\overline{L} = \int_{Γ^-}L^+(γ^-)d μ_{\mathrm{inv}}(γ^-)$.
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