Title: On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum
Author(s):

	E.Lytvynov (Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K.) ,
	P.T.Polara (Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K.)

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 L²-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.

Key words: birth-and-death process, continuous system, Gibbs measure, hopping particles, scaling limit
PACS: 02.50.Ey, 02.50.Ga