Diffusion approximation for equilibrium Kawasaki dynamics in continuum

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:118

Article

Kondratiev, Yuri G.^2,3,4  Kutoviy, Oleksandr V.^3,4  Lytvynovd, Eugene W.¹ 

[1] Univ Coll Swansea, Dept Math, Swansea SA2 8PP, W Glam, Wales

[2] Kiev Mohyla Acad, Kiev, Ukraine

[3] Univ Bielefeld, BiBoS, D-4800 Bielefeld, Germany

[4] Univ Bielefeld, Fak Math, D-33501 Bielefeld, Germany

关键词: continuous system;    diffusion approximation;    Gibbs measure;    gradient stochastic dynamics;    Kawasaki dynamics in continuum;    scaling limit;   

DOI  :  10.1016/j.spa.2007.09.001

来源: Elsevier

【 摘 要 】

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 mu 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, phi, (in particular, admitting a singularity of phi 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 phi is from C(b)(3) (R(d)) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [V. Choi, Y.M. Park, H.J. Yoo, Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness, J. Math. Phys. 39 (1998) 6509-6536]. (C) 2007 Elsevier B.V. All rights reserved.

【 授权许可】

Free