Glauber dynamics of continuous particle systems
Glauber dynamics of continuous particle systems
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DOI：
10.1016/j.anihpb.2004.05.002
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发表时间：
2003-06
影响因子：
1.5
通讯作者：
Yuri Kondratiev;Eugene Lytvynov;Eugene Lytvynov
中科院分区：
文献类型：
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作者：
Yuri Kondratiev;Eugene Lytvynov;Eugene Lytvynov
本文构造和研究了无限连续质点系统的平衡Glauber型动力学。这种动力学是空间生灭过程的一个特例。在Rd中所有局部有限子集（组态）的空间Γ上，我们固定一个对应于一般对势φ和活度z> 0的吉布斯测度μ。我们考虑L2（Γ，μ）上的Dirichlet型E，它对应于Glauber动力学的生成元H.本文证明了与E恰当相关的马氏过程M在Γ上的存在性。在正势φ满足δ：=<$Rd（1− e− φ（x））zdx< 1的情况下，我们还证明了生成元H有一个谱隙<$1 − δ。此外，对于任何纯吉布斯态μ，我们导出一个庞加莱不等式。关于谱隙和Poincaré不等式的结果是[Ann.Inst.H.庞加莱概率中央集权主义者38（2002）91-108]. All rights reserved.
This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ, µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ:=∫ Rd (1− e− φ (x)) zdx< 1, we also prove that the generator H has a spectral gap⩾ 1− δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 91–108]. 2004 Elsevier SAS. All rights reserved.