Diffusion approximation for equilibrium Kawasaki dynamics in continuum

2007/02/07 by Y. G. Kondratiev, Kondratiev, Y. G., O. V. Kutoviy +3
Mathematics · Physics and Astronomy · #60F99 #60J60 #60J75 #60K35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60F99 #msc:60J60 #msc:60J75 #msc:60K35

paper · pdf · doi:10.48550/arxiv.math/0702178

arxiv created 2007/08/20 · arxiv updated 2009/12/01

Abstract

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in \mathbb 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_{\mathrm b}³(\mathbb R^d) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi \it et al., J. Math. Phys. 39 (1998) 6509--6536].

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