{"title":"Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential","authors":"Martin Grothaus, Simon Wittmann","doi":"10.1007/s00020-024-02775-6","DOIUrl":null,"url":null,"abstract":"<p>This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, <span>\\({\\mathcal {E}}^N\\)</span> on <span>\\(L^2(E,\\mu _N)\\)</span> for <span>\\(N\\in {\\mathbb {N}}\\)</span>, in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family <span>\\({(\\mu _N)}_{N}\\)</span> on the state space <i>E</i>—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on <span>\\({(\\mu _N)}_{N}\\)</span> try to impose as little restrictions as possible. The problem has fully been solved if the family <span>\\({(\\mu _N)}_{N}\\)</span> contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).\n</p>","PeriodicalId":13658,"journal":{"name":"Integral Equations and Operator Theory","volume":"17 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Integral Equations and Operator Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00020-024-02775-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, \({\mathcal {E}}^N\) on \(L^2(E,\mu _N)\) for \(N\in {\mathbb {N}}\), in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family \({(\mu _N)}_{N}\) on the state space E—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on \({(\mu _N)}_{N}\) try to impose as little restrictions as possible. The problem has fully been solved if the family \({(\mu _N)}_{N}\) contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).
期刊介绍:
Integral Equations and Operator Theory (IEOT) is devoted to the publication of current research in integral equations, operator theory and related topics with emphasis on the linear aspects of the theory. The journal reports on the full scope of current developments from abstract theory to numerical methods and applications to analysis, physics, mechanics, engineering and others. The journal consists of two sections: a main section consisting of refereed papers and a second consisting of short announcements of important results, open problems, information, etc.