AI 中文总结
本文提出一种通用准控制近似方法，通过装饰树、迭代并行积提升等技术，结合BPHZ重整化，建立抛物型奇异随机偏微分方程的通用解理论。
AI 中文摘要
本文为Gubinelli、Imkeller和Perkowski提出的用于处理奇异随机偏微分方程（singular SPDEs）的准控制方法提供了一种通用近似方法，该近似方法涵盖了一大类方程，通过编码其系数和随机数据的装饰树进行描述。主要创新之处在于准控制随机迭代积分的递归定义，该积分涉及精心选择的组合词集；平行化通过将迭代 paraproducts（并行积）提升到模型分布，并结合来自正则结构（Regularity Structures）的重构定理来实现。当使用包含BPHZ重整化的制备映射形式时，本文还提供了基于该近似方法的不动点论证和重整化方程。最后一个主要结果是，通过调整谱间隙方法，利用BPHZ重整化的重整化随机数据收敛。最终，通过准控制方法获得了抛物型奇异随机偏微分方程的通用解理论。
英文摘要
We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions and the reconstruction theorem coming from Regularity Structures. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormsalition. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation via an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.
Comments106 pages
