Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential

IF 0.8 3区 数学 Q2 MATHEMATICS
Martin Grothaus, Simon Wittmann
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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).

Abstract Image

具有非凸相互作用势的梯度形式的莫斯科收敛性
本文在K. Kuwae和T. Shioya的收敛希尔伯特空间框架下,提供了一种新的方法来解决梯度型狄利克雷形式的Mosco收敛问题,即\({\mathcal {E}}^N\) on \(L^2(E,\mu _N)\) for \(N\in {\mathbb {N}}\).基本假设是状态空间 E--可分离的希尔伯特空间或局部凸拓扑向量空间--上的族\({(\mu _N)}_{N}\)的弱度量收敛。除此之外,\({(\mu _N)}_{N}\) 上的条件尽量少施加限制。如果族 \({(\mu_N)}_{N}\)只包含对数凹计量,那么这个问题就完全解决了,这归功于 Ambrosio 等人的研究(Probab Theory Relat. Fields 145:517-564, 2009)。然而,对于一大类收敛问题,对数凹性假设失效了。这篇文章提出了一种克服这一障碍的新方法。结合狄利克特形式理论和数值分析方法,我们找到了具有不同参考量的标准梯度形式的 Mosco 收敛的抽象标准。其中包括测量值不是对数凹的情况。为了证明我们的抽象理论的易用性,我们讨论了第一个应用,概括了 Bounebache 和 Zambotti 的近似结果(J Theor Probab 27:168-201, 2014)。
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来源期刊
CiteScore
1.60
自引率
0.00%
发文量
36
审稿时长
6 months
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.
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