Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum

摘要

The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein-Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential phi only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition (I) and a conjecture on the Percus-Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have cadlag sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential. (c) 2006 Elsevier Inc. All rights reserved.