Dirichlet form approach to interacting particle systems with long range interactions on $mathbb{Z}^{d}$ (Stochastic Analysis on Large Scale Interacting Systems)

Abstract

In this paper we present a general theorem of constructing interacting particle systems with long range interactions on discrete spaces. It can be applied to the system that interaction between particles is given by the logarithmic potential. If its equilibrium measure μ is translation invariant we can construct the system whose particles have a summable jump rate. In addition the decay order of jump rate is restricted by the growth order of the 1-correlation function of the measure μ in general cases. The results are the discrete counter part of the results in [2]. In addition we discuss Glauber dynamics whose equilibrium measures are associated with these long range interactions.