{"title":"Homogenization of convolution type semigroups in high contrast media","authors":"Andrey Piatnitski, Elena Zhizhina","doi":"10.1007/s13324-025-01034-0","DOIUrl":null,"url":null,"abstract":"<div><p>The paper deals with the asymptotic properties of semigroups associated with Markov jump processes in a high contrast periodic medium in <span>\\(\\mathbb {R}^d\\)</span>, <span>\\(d\\ge 1\\)</span>. In order to study the limit behaviour of these semigroups we equip the corresponding Markov processes with an extra variable that characterizes the position of the process inside the period, and show that the limit dynamics of these two-component processes remains Markov. We describe the limit process and prove the convergence of the corresponding semigroups as well as the convergence in law of the extended processes in the path space. Since the components of the limit process are coupled, the dynamics of the first (spacial) component need not have a semigroup property. We derive the evolution equation with a memory term for the dynamics of this component of the limit process. We also discuss the construction of the limit semigroup in the <span>\\(L^2\\)</span> space and study the spectrum of its generator.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"15 2","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13324-025-01034-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-025-01034-0","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: 0
Abstract
The paper deals with the asymptotic properties of semigroups associated with Markov jump processes in a high contrast periodic medium in \(\mathbb {R}^d\), \(d\ge 1\). In order to study the limit behaviour of these semigroups we equip the corresponding Markov processes with an extra variable that characterizes the position of the process inside the period, and show that the limit dynamics of these two-component processes remains Markov. We describe the limit process and prove the convergence of the corresponding semigroups as well as the convergence in law of the extended processes in the path space. Since the components of the limit process are coupled, the dynamics of the first (spacial) component need not have a semigroup property. We derive the evolution equation with a memory term for the dynamics of this component of the limit process. We also discuss the construction of the limit semigroup in the \(L^2\) space and study the spectrum of its generator.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.