Essential spectral radius for Markov semigroups (I): discrete time case

• 成果类型: Article
• 署名作者: Wu, LM
• 署名单位: Wuhan University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-003-0304-0
• 发表日期: 2004
• 页码: 255-321
• 关键词: large deviations; functional inequalities; Moderate Deviations; Operators; gap; CONVERGENCE

摘要

Using two new measures of non-compactness beta(tau) (P) and beta(w) (P) for a positive kernel P on a Polish space E, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius r(ess) (P) on bB. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblin's condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space b(u)B. Some related results on L-2 (mu) are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero; this result allows us to show that the sufficient condition of Donsker-Varadhan for the large deviation principle is in fact necessary. The knowledge of r(ess) (P) allows us to estimate eigenvalues of P in L-2 in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Applications to different concrete models are provided for illustrating those general results.