On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

TL;DR

This paper proves that Glauber dynamics can be obtained as a scaling limit of Kawasaki dynamics for infinite particle systems in continuum, establishing convergence of generators in the $L^2$-norm under Gibbs measures.

Contribution

It demonstrates the convergence of generators of Kawasaki dynamics to those of Glauber dynamics in continuum, generalizing previous specific cases.

Findings

Glauber dynamics derived as a scaling limit of Kawasaki dynamics

Convergence of generators established in $L^2$-norm

Applicable to Gibbs measures in low activity-high temperature regime

Abstract

We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics.