Diffusion approximation for equilibrium Kawasaki dynamics in continuum

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in RdRd which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure μμ 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, ϕϕ, (in particular, admitting a singularity of ϕϕ 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 ϕϕ is from Cb3(Rd) 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].