{"title":"The Brownian Web as a random R-tree","authors":"G. Cannizzaro, Martin Hairer","doi":"10.1214/23-ejp984","DOIUrl":null,"url":null,"abstract":"Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\\mathbb R^2$, as a random variable taking values in the space of (spatial) $\\mathbb R$-trees. This gives a stronger topology than the classical one {(i.e.\\ Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $\\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g.\\ its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/23-ejp984","RegionNum":3,"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
Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\mathbb R^2$, as a random variable taking values in the space of (spatial) $\mathbb R$-trees. This gives a stronger topology than the classical one {(i.e.\ Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g.\ its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.
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.