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 $\Gamma^+ \times \Gamma^-$ 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 $L^+(\gamma^-) + \frac{1}{\varepsilon}L^-$, $\varepsilon > 0$. Here $L^-$ describes the environment process on $\Gamma^-$ and $L^+(\gamma^-)$ describes the system process on $\Gamma^+$, where $\gamma^-$ indicates that the corresponding birth-and-death rates depend on another locally finite configuration $\gamma^- \in \Gamma^-$. 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 $\mu_t^{\varepsilon}$ on $\Gamma^+ \times \Gamma^-$. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let $\mu_{\mathrm{inv}}$ be the invariant measure for the environment process on $\Gamma^-$. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of $\mu_t^{\varepsilon}$ onto $\Gamma^+$ converges weakly to an evolution of states on $\Gamma^+$ associated with the averaged Markov birth-and-death operator $\overline{L} = \int_{\Gamma^-}L^+(\gamma^-)d \mu_{\mathrm{inv}}(\gamma^-)$.