{"title":"Functional inequalities for transient birth–death processes and their applications","authors":"Jian Wang","doi":"10.1016/j.jmaa.2009.10.075","DOIUrl":null,"url":null,"abstract":"<div><p>We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth–death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth–death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.</p></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"364 1","pages":"Pages 171-185"},"PeriodicalIF":1.2000,"publicationDate":"2010-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jmaa.2009.10.075","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X09009299","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth–death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth–death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.
期刊介绍：
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
Papers are sought which employ one or more of the following areas of classical analysis:
• Analytic number theory
• Functional analysis and operator theory
• Real and harmonic analysis
• Complex analysis
• Numerical analysis
• Applied mathematics
• Partial differential equations
• Dynamical systems
• Control and Optimization
• Probability
• Mathematical biology
• Combinatorics
• Mathematical physics.
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