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 Γ{sup +}×Γ{sup −} over ℝ{sup d}. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L{sup +}(γ{sup −})+(1/ε)L{sup −}, ε>0. Here L{sup −} describes the environment process on Γ{sup −} and L{sup +}(γ{sup −}) describes the system process on Γ{sup +}, where γ{sup −} indicates that the corresponding birth-and-death rates depend on another locally finite configuration γ{sup −}∈Γ{sup −}. 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{sub t}{sup ε} on Γ{sup +}×Γ{sup −}. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let μ{sub inv} be the invariant measure for the environment process on Γ{sup −}. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of μ{sub t}{sup ε} onto Γ{sup +} converges weakly to an evolution of states on Γ{sup +} associated with the averaged Markov birth-and-death operator L̄=∫{sub Γ{sup −}}L{sup +}(γ{sup −})dμ{sub inv}(γ{sup −}).