Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations ({varGamma }^+ times {varGamma }^-) 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 ({varGamma }^-) and (L^+(gamma ^-)) describes the system process on ({varGamma }^+), where (gamma ^-) indicates that the corresponding birth-and-death rates depend on another locally finite configuration (gamma ^- in {varGamma }^-). 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 ({varGamma }^+ times {varGamma }^-). 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 ({varGamma }^-). 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 ({varGamma }^+) converges weakly to an evolution of states on ({varGamma }^+) associated with the averaged Markov birth-and-death operator ({overline{L}} = int _{{varGamma }^-}L^+(gamma ^-)d mu _{mathrm {inv}}(gamma ^-)).