Binary jumps in continuum. I. Equilibrium processes and their scaling limits
Let Gamma denote the space of all locally finite subsets (configurations) in R(d). A stochastic dynamics of binary jumps in continuum is a Markov process on Gamma in which pairs of particles simultaneously hop over R(d). In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birth-and-death process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3601118]
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American Institute of Physics (AIP)