{"title":"Brownian Particle in the Curl of 2-D Stochastic Heat Equations","authors":"Guilherme de Lima Feltes, Hendrik Weber","doi":"10.1007/s10955-023-03224-1","DOIUrl":null,"url":null,"abstract":"<p>We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp <span>\\(\\sqrt{\\log }\\)</span>-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) <span>\\(\\underline{\\omega }\\)</span>. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of <span>\\(\\underline{\\omega }\\)</span>. Adapting their method, we show that if <span>\\(s\\ge 1\\)</span>, with <span>\\(s=1\\)</span> corresponding to the standard stochastic heat equation, then the particle stays <span>\\(\\sqrt{\\log }\\)</span>-super diffusive, whereas if <span>\\(s<1\\)</span>, corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for <span>\\(s<1\\)</span>, we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the <span>\\(\\log \\)</span>-corrections to diffusivity. We also provide examples which display <span>\\(\\log ^a\\)</span>-super diffusive behaviour for <span>\\(a\\in (0,1/2]\\)</span>.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-01-28","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/s10955-023-03224-1","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}
引用次数: 0
Abstract
We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp \(\sqrt{\log }\)-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) \(\underline{\omega }\). We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of \(\underline{\omega }\). Adapting their method, we show that if \(s\ge 1\), with \(s=1\) corresponding to the standard stochastic heat equation, then the particle stays \(\sqrt{\log }\)-super diffusive, whereas if \(s<1\), corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for \(s<1\), we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the \(\log \)-corrections to diffusivity. We also provide examples which display \(\log ^a\)-super diffusive behaviour for \(a\in (0,1/2]\).
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.