Doob equivalence and non-commutative peaking for Markov chains

IF 0.7 2区 数学 Q2 MATHEMATICS Journal of Noncommutative Geometry Pub Date : 2019-11-23 DOI:10.4171/jncg/444
Xinxin Chen, Adam Dor-On, Langwen Hui, C. Linden, Yifan Zhang
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引用次数: 5

Abstract

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.
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马尔可夫链的Doob等价与非交换峰
本文给出了由随机矩阵构造算子代数的问题如何在马尔可夫链上调和函数的研究中激发出新的结果。更准确地说,我们用(广义的)Doob变换来描述条件概率的重合,这就导致了相关算子代数在谱半径和强刘维尔性质方面的更强分类结果。此外,我们以一种允许人们通过检查矩阵来确定它们的方式来表征关联算子代数的非交换峰值点。这导致了计算放大算子代数中算子范数的最大模原理的具体模拟。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.60
自引率
11.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: The Journal of Noncommutative Geometry covers the noncommutative world in all its aspects. It is devoted to publication of research articles which represent major advances in the area of noncommutative geometry and its applications to other fields of mathematics and theoretical physics. Topics covered include in particular: Hochschild and cyclic cohomology K-theory and index theory Measure theory and topology of noncommutative spaces, operator algebras Spectral geometry of noncommutative spaces Noncommutative algebraic geometry Hopf algebras and quantum groups Foliations, groupoids, stacks, gerbes Deformations and quantization Noncommutative spaces in number theory and arithmetic geometry Noncommutative geometry in physics: QFT, renormalization, gauge theory, string theory, gravity, mirror symmetry, solid state physics, statistical mechanics.
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