分数 $$\Phi ^3_d$$ 模型的 BPHZ 重正化和消失次临界渐近性

IF 1.4 3区 数学 Q2 MATHEMATICS, APPLIED
Nils Berglund, Yvain Bruned
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引用次数: 0

摘要

我们考虑了d维环面上的随机PDEs,它们具有参数为(\rho \in (0,2]\)的分数拉普拉斯,二次非线性,并由时空白噪声驱动。众所周知,这些方程是局部次临界的,因此适用于正则结构理论,当且仅当\(\rho > d/3\)时。利用第二作者、A. Chandra、I. Chevyrev、M. Hairer和L. Zambotti的一系列最新成果,我们得到了当mollification参数\(\varepsilon \)变小且\(\rho \)接近临界值时正则化反求的精确渐近线。特别是,我们证明如果 \(\varepsilon \) 在 \((\rho -d/3)\) 中是超指数小的,那么反项则像\(\log (\varepsilon ^{-1})\)的负幂次,反之则是\(\log (\varepsilon ^{-1})\)。这项工作也是在相对简单的情况下对BPHZ重正化一般理论的说明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $$\Phi ^3_d$$ model

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.

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来源期刊
CiteScore
2.70
自引率
13.30%
发文量
54
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.
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