{"title":"Expected hitting time estimates on finite graphs","authors":"Laurent Saloff-Coste , Yuwen Wang","doi":"10.1016/j.spa.2025.104626","DOIUrl":null,"url":null,"abstract":"<div><div>The expected hitting time from vertex <span><math><mi>a</mi></math></span> to vertex <span><math><mi>b</mi></math></span>, <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span>, is the expected value of the time it takes a random walk starting at <span><math><mi>a</mi></math></span> to reach <span><math><mi>b</mi></math></span>. In this paper, we give estimates for <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> when the distance between <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span> is comparable to the diameter of the graph, and the graph satisfies a Harnack condition. We show that, in such cases, <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> can be estimated in terms of the volumes of balls around <span><math><mi>b</mi></math></span>. Using our results, we estimate <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math></span> on various graphs, such as rectangular tori, some convex traces in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and fractal graphs. Our proofs use heat kernel estimates.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"185 ","pages":"Article 104626"},"PeriodicalIF":1.1000,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414925000675","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
The expected hitting time from vertex to vertex , , is the expected value of the time it takes a random walk starting at to reach . In this paper, we give estimates for when the distance between and is comparable to the diameter of the graph, and the graph satisfies a Harnack condition. We show that, in such cases, can be estimated in terms of the volumes of balls around . Using our results, we estimate on various graphs, such as rectangular tori, some convex traces in , and fractal graphs. Our proofs use heat kernel estimates.
期刊介绍:
Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests.
Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.