{"title":"平滑自相似异常扩散过程的持续概率","authors":"Frank Aurzada, Pascal Mittenbühler","doi":"10.1007/s10955-024-03251-6","DOIUrl":null,"url":null,"abstract":"<p>We consider the persistence probability of a certain fractional Gaussian process <span>\\(M^H\\)</span> that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of <span>\\(M^H\\)</span> exists, is positive and continuous in the Hurst parameter <i>H</i>. Further, the asymptotic behaviour of the persistence exponent for <span>\\(H\\downarrow 0\\)</span> and <span>\\(H\\uparrow 1\\)</span>, respectively, is studied. Finally, for <span>\\(H\\rightarrow 1/2\\)</span>, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that <span>\\(M^{1/2}\\)</span> vanishes.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process\",\"authors\":\"Frank Aurzada, Pascal Mittenbühler\",\"doi\":\"10.1007/s10955-024-03251-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the persistence probability of a certain fractional Gaussian process <span>\\\\(M^H\\\\)</span> that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of <span>\\\\(M^H\\\\)</span> exists, is positive and continuous in the Hurst parameter <i>H</i>. Further, the asymptotic behaviour of the persistence exponent for <span>\\\\(H\\\\downarrow 0\\\\)</span> and <span>\\\\(H\\\\uparrow 1\\\\)</span>, respectively, is studied. Finally, for <span>\\\\(H\\\\rightarrow 1/2\\\\)</span>, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that <span>\\\\(M^{1/2}\\\\)</span> vanishes.</p>\",\"PeriodicalId\":667,\"journal\":{\"name\":\"Journal of Statistical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-03-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Statistical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s10955-024-03251-6\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03251-6","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Persistence Probabilities of a Smooth Self-Similar Anomalous Diffusion Process
We consider the persistence probability of a certain fractional Gaussian process \(M^H\) that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of \(M^H\) exists, is positive and continuous in the Hurst parameter H. Further, the asymptotic behaviour of the persistence exponent for \(H\downarrow 0\) and \(H\uparrow 1\), respectively, is studied. Finally, for \(H\rightarrow 1/2\), the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that \(M^{1/2}\) vanishes.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.