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

Lytvynov Lytvynov,Polara Polara

doi:10.5488/cmp.11.2.223

Abstract

We deal with the 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 [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.

Highlights

This paper deals with two classes of equilibrium stochastic dynamics of infinite particle systems in continuum
Birth-and-death processes in continuum (Glauber dynamics), i.e., dynamics where there is no motion of particles, but rather particles disappear or appear at random, see, e.g., [1,6,8,11,13,14,19,23];
In [3], it was proved that a special Glauber dynamics can be derived through a scaling limit of Kawasaki dynamics

Summary

Introduction

This paper deals with two classes of equilibrium stochastic dynamics of infinite particle systems in continuum. In [3], it was proved that a special Glauber dynamics can be derived through a scaling limit of Kawasaki dynamics It was conjectured in [3] that such a result holds, for a wide class of birth-and-death dynamics (dynamics of hopping particles, respectively), which are indexed by a parameter s ∈ [0, 1]. (In the case where s ∈ (1/2, 1], one needs to put additional, quite restrictive assumptions on the potential of pair interaction, and we will not treat this case in the present paper.) we show that the result of [3] is not a property of just one special Kawasaki (Glauber, respectively) dynamics, but rather represents a property which is common to many dynamics. The authors acknowledge numerous useful discussions with Dmirti Finkelshtein and Yuri Kondratiev

Gibbs measures in the low activity-high temperature regime

Scaling limit