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

Friesen M, Kondratiev Y. Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS. 2018;171(5):842-877.We study a spatial birth-and-death process on the phase space of locally finite configurations Gamma(+) x Gamma(-) over 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(-)) + 1/epsilon L-, epsilon > 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(-) is an element of 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(epsilon)(t) on Gamma(+) x Gamma(-). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let mu(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(epsilon)(t) onto Gamma(+) converges weakly to an evolution of states on Gamma(+) associated with the averaged Markov birth-and-death operator (L) over bar = integral(Gamma)-L+(gamma(-))D mu(inv) (gamma(-))