Binary jumps in continuum. I. Equilibrium processes and their scaling limits

Finkelshtein DL, Kondratiev Y, Kutoviy OV, Lytvynov E. Binary jumps in continuum. I. Equilibrium processes and their scaling limits. Journal of Mathematical Physics. 2011;52(6): 063304.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