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

Abstract

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 ${\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)$.