{"title":"二维厚点的紧密性","authors":"J. Rosen","doi":"10.1214/23-ejp910","DOIUrl":null,"url":null,"abstract":"Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \\theta$ be the first exit time of the unit disk $D_{1}$. Let \\[\\mu_{ \\theta } ( x,\\epsilon) =\\frac{1}{\\pi\\epsilon^{ 2} }\\int_{0}^{ \\theta }1_{\\{ B( x,\\epsilon)\\}}( W_{t})\\,dt,\\] and set $\\mu^{ \\ast}_{ \\theta } (\\epsilon)=\\sup_{x\\in D_{1}}\\mu_{ \\theta } ( x,\\epsilon)$. We show that \\[\\sqrt{\\mu^{\\ast}_{\\theta} (\\epsilon)}-\\sqrt{2/\\pi} \\left(\\log \\epsilon^{-1}- \\frac{1}{2}\\log\\log \\epsilon^{-1}\\right)\\] is tight.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Tightness for thick points in two dimensions\",\"authors\":\"J. Rosen\",\"doi\":\"10.1214/23-ejp910\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \\\\theta$ be the first exit time of the unit disk $D_{1}$. Let \\\\[\\\\mu_{ \\\\theta } ( x,\\\\epsilon) =\\\\frac{1}{\\\\pi\\\\epsilon^{ 2} }\\\\int_{0}^{ \\\\theta }1_{\\\\{ B( x,\\\\epsilon)\\\\}}( W_{t})\\\\,dt,\\\\] and set $\\\\mu^{ \\\\ast}_{ \\\\theta } (\\\\epsilon)=\\\\sup_{x\\\\in D_{1}}\\\\mu_{ \\\\theta } ( x,\\\\epsilon)$. We show that \\\\[\\\\sqrt{\\\\mu^{\\\\ast}_{\\\\theta} (\\\\epsilon)}-\\\\sqrt{2/\\\\pi} \\\\left(\\\\log \\\\epsilon^{-1}- \\\\frac{1}{2}\\\\log\\\\log \\\\epsilon^{-1}\\\\right)\\\\] is tight.\",\"PeriodicalId\":50538,\"journal\":{\"name\":\"Electronic Journal of Probability\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2022-03-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1214/23-ejp910\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/23-ejp910","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \theta$ be the first exit time of the unit disk $D_{1}$. Let \[\mu_{ \theta } ( x,\epsilon) =\frac{1}{\pi\epsilon^{ 2} }\int_{0}^{ \theta }1_{\{ B( x,\epsilon)\}}( W_{t})\,dt,\] and set $\mu^{ \ast}_{ \theta } (\epsilon)=\sup_{x\in D_{1}}\mu_{ \theta } ( x,\epsilon)$. We show that \[\sqrt{\mu^{\ast}_{\theta} (\epsilon)}-\sqrt{2/\pi} \left(\log \epsilon^{-1}- \frac{1}{2}\log\log \epsilon^{-1}\right)\] is tight.
期刊介绍:
The Electronic Journal of Probability publishes full-size research articles in probability theory. The Electronic Communications in Probability (ECP), a sister journal of EJP, publishes short notes and research announcements in probability theory.
Both ECP and EJP are official journals of the Institute of Mathematical Statistics
and the Bernoulli society.