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Stochastic averaging principle for spatial Markov evolutions in the continuum

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 $\overline{L} = \int_{Γ^-}L^+(γ^-)d μ_{\mathrm{inv}}(γ^-)$.