On the singular values of complex matrix Brownian motion with a matrix drift

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2021-07-11 DOI:10.3150/22-bej1517
T. Assiotis
{"title":"On the singular values of complex matrix Brownian motion with a matrix drift","authors":"T. Assiotis","doi":"10.3150/22-bej1517","DOIUrl":null,"url":null,"abstract":"Let $Mat_{\\mathbb{C}}(K,N)$ be the space of $K\\times N$ complex matrices. Let $\\mathbf{B}_t$ be Brownian motion on $Mat_{\\mathbb{C}}(K,N)$ starting from the zero matrix and $\\mathbf{M}\\in Mat_{\\mathbb{C}}(K,N)$. We prove that, with $K\\ge N$, the $N$ eigenvalues of $\\left(\\mathbf{B}_t+t\\mathbf{M}\\right)^*\\left(\\mathbf{B}_t+t\\mathbf{M}\\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bernoulli","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/22-bej1517","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1

Abstract

Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let $\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the $N$ eigenvalues of $\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.
具有矩阵漂移的复矩阵Brownian运动的奇异值
设$Mat_{\mathbb{C}}(K,N)$是$K\timesN$复矩阵的空间。让$\mathbf{B}_t$be从零矩阵开始的$Mat_{\mathbb{C}}(K,N)$上的Brownian运动和Mat_{\mathbb{C}}(K,N)$中的$\mathbf{M}\。我们证明了在$K\geN$的情况下,$\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$形成具有显式转换核的马尔可夫过程。这推广了Rogers和Pitman关于具有漂移的多维布朗运动的一个经典结果,该结果对应于$N=1$。然后,我们对这个马尔可夫过程又给出了两个描述。首先,作为广义的独立平方贝塞尔扩散过程,由渡边介绍,Pitman和Yor研究,条件是永远不相交。第二,作为一些交错阵列中相互作用的贝塞尔型扩散的顶行的分布。最后两个描述也扩展到一类一般的一维扩散。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信