{"title":"分数 $$\\Phi ^3_d$$ 模型的 BPHZ 重正化和消失次临界渐近性","authors":"Nils Berglund, Yvain Bruned","doi":"10.1007/s40072-024-00331-2","DOIUrl":null,"url":null,"abstract":"<p>We consider stochastic PDEs on the <i>d</i>-dimensional torus with fractional Laplacian of parameter <span>\\(\\rho \\in (0,2]\\)</span>, quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if <span>\\(\\rho > d/3\\)</span>. Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter <span>\\(\\varepsilon \\)</span> becomes small and <span>\\(\\rho \\)</span> approaches its critical value. In particular, we show that the counterterms behave like a negative power of <span>\\(\\varepsilon \\)</span> if <span>\\(\\varepsilon \\)</span> is superexponentially small in <span>\\((\\rho -d/3)\\)</span>, and are otherwise of order <span>\\(\\log (\\varepsilon ^{-1})\\)</span>. This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.</p>","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $$\\\\Phi ^3_d$$ model\",\"authors\":\"Nils Berglund, Yvain Bruned\",\"doi\":\"10.1007/s40072-024-00331-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider stochastic PDEs on the <i>d</i>-dimensional torus with fractional Laplacian of parameter <span>\\\\(\\\\rho \\\\in (0,2]\\\\)</span>, quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if <span>\\\\(\\\\rho > d/3\\\\)</span>. Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter <span>\\\\(\\\\varepsilon \\\\)</span> becomes small and <span>\\\\(\\\\rho \\\\)</span> approaches its critical value. In particular, we show that the counterterms behave like a negative power of <span>\\\\(\\\\varepsilon \\\\)</span> if <span>\\\\(\\\\varepsilon \\\\)</span> is superexponentially small in <span>\\\\((\\\\rho -d/3)\\\\)</span>, and are otherwise of order <span>\\\\(\\\\log (\\\\varepsilon ^{-1})\\\\)</span>. This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.</p>\",\"PeriodicalId\":48569,\"journal\":{\"name\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-06-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Stochastics and Partial Differential Equations-Analysis and Computations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s40072-024-00331-2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics and Partial Differential Equations-Analysis and Computations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40072-024-00331-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $$\Phi ^3_d$$ model
We consider stochastic PDEs on the d-dimensional torus with fractional Laplacian of parameter \(\rho \in (0,2]\), quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if \(\rho > d/3\). Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter \(\varepsilon \) becomes small and \(\rho \) approaches its critical value. In particular, we show that the counterterms behave like a negative power of \(\varepsilon \) if \(\varepsilon \) is superexponentially small in \((\rho -d/3)\), and are otherwise of order \(\log (\varepsilon ^{-1})\). This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.
期刊介绍:
Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.