{"title":"Study of a fractional stochastic heat equation","authors":"N. Schaeffer","doi":"10.30757/alea.v20-15","DOIUrl":null,"url":null,"abstract":"In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: \\begin{equation*} \\left\\{\\begin{array}{l} \\partial_t u-\\Delta u= \\rho^2 u^2 + \\dot B \\, , \\quad t\\in [0,T] \\, , \\, x\\in \\mathbb{R}^d \\, ,\\\\ u_0=\\phi\\, . \\end{array} \\right. \\end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $\\in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+\\sum_{i=1}^{d}H_i>d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+\\sum_{i=1}^{d}H_i \\leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $d\\geq1.$","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v20-15","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1
Abstract
In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: \begin{equation*} \left\{\begin{array}{l} \partial_t u-\Delta u= \rho^2 u^2 + \dot B \, , \quad t\in [0,T] \, , \, x\in \mathbb{R}^d \, ,\\ u_0=\phi\, . \end{array} \right. \end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $\in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+\sum_{i=1}^{d}H_i>d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+\sum_{i=1}^{d}H_i \leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $d\geq1.$
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.