Logarithmic Sobolev inequality for generalized simple exclusion processes

• 成果类型: Article
• 署名作者: Yau, HT
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050140
• 发表日期: 1997
• 页码: 507-538
• 关键词: glauber dynamics

摘要

Let <(mu)over bar> be a probability measure on the set {0,1,...,R} for some R is an element of N and Lambda(L) a cube of width L in Z(d). Denote by mu(Lambda L)(gc) the (grand canonical) product measure on the configuration space on Lambda(L) with <(mu)over bar> as the marginal measure; here the superscript indicates the grand canonical ensemble. The canonical ensemble, denoted by mu(Lambda L,n)(c), is defined by conditioning mu(Lambda L)(gc) given the total number of particles to be n. Consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site. The rates are chosen such that, for every n and L fixed, the measure mu(Lambda L,n)(c) is reversible. We prove the logarithmic Sobolev inequality in the sense that integral f log f d mu(Lambda L,n)(c) less than or equal to const. (LD)-D-2(root f) for any probability density f with respect to mu(Lambda L,n)(c); here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal.