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Diffusion approximation for equilibrium Kawasaki dynamics in continuum (2007)
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BibTeX
@MISC{Kondratiev07diffusionapproximation,
author = {Yuri G. Kondratiev and Oleksandr V. Kutoviy and Eugene W. Lytvynov},
title = {Diffusion approximation for equilibrium Kawasaki dynamics in continuum},
year = {2007}
}
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Abstract
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, φ, (in particular, admitting a singularity of φ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential φ is from C 3 b (Rd) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi et al., J. Math.
Keyphrases
equilibrium kawasaki dynamic diffusion approximation smooth local function diffusion generator corresponding equilibrium finite-dimensional distribution jump rate weak convergence invariant measure gradient stochastic dynamic pair interaction gibbs measure kawasaki dynamic scaled dynamic converges infinite system diffusive limit latter result weak assumption
