Diffusion approximation for equilibrium Kawasaki dynamics in continuum

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 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].