Naomi D. Feldheim, Ohad N. Feldheim, Sumit Mukherjee
{"title":"Persistence and ball exponents for Gaussian stationary processes","authors":"Naomi D. Feldheim, Ohad N. Feldheim, Sumit Mukherjee","doi":"10.1002/cpa.22255","DOIUrl":null,"url":null,"abstract":"Consider a real Gaussian stationary process , indexed on either or and admitting a spectral measure . We study , the persistence exponent of . We show that, if has a positive density at the origin, then the persistence exponent exists; moreover, if has an absolutely continuous component, then if and only if this spectral density at the origin is finite. We further establish continuity of in , in (under a suitable metric) and, if is compactly supported, also in dense sampling. Analogous continuity properties are shown for , the ball exponent of , and it is shown to be positive if and only if has an absolutely continuous component.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"222 1","pages":""},"PeriodicalIF":3.1000,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/cpa.22255","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Consider a real Gaussian stationary process , indexed on either or and admitting a spectral measure . We study , the persistence exponent of . We show that, if has a positive density at the origin, then the persistence exponent exists; moreover, if has an absolutely continuous component, then if and only if this spectral density at the origin is finite. We further establish continuity of in , in (under a suitable metric) and, if is compactly supported, also in dense sampling. Analogous continuity properties are shown for , the ball exponent of , and it is shown to be positive if and only if has an absolutely continuous component.