[论文解读] An analytic BPHZ theorem for regularity structures

Ajay Chandra, Martin Hairer|arXiv (Cornell University)|Dec 24, 2016

Stochastic processes and financial applications参考文献 22被引用 103

一句话总结

本文证明了一个在正则性结构框架内用于非线性SPDEs的再正规化模型随机收敛性的通用、自动化框架，能够以黑箱方式产生估计，并允许非高斯驱动。

ABSTRACT

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Hölder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.

研究动机与目标

• 在正则性结构框架内激发对普遍随机收敛定理的需求。
• 提供一种黑箱方法，能够自动为广泛类别的非线性SPDEs产生随机估计。
• 结合多尺度分析，并解决来自在正则性结构中重新居中引起的正性重整化。
• 处理非高斯驱动场以及重整化群作用之间的差异。
• 证明在某些Hölder空间上，canonical升不连续，而在适当条件下，BPHZ升在分布意义上连续。

提出的方法

• 构建一种多尺度分析框架，类似于构造性场论。
• 将正性重整化从正则性结构中的重新居中引入。
• 利用通过指定公式 [BHZalg] 计算的BPHZ重整化常数。
• 在重整化方案中处理非高斯驱动场。
• 证明在具有足够矩的平稳随机场中，BPHZ升的分布连续性。

实验结果

研究问题

• RQ1 是否存在一个普遍的、自动化的定理来保证在正则性结构内非线性SPDEs的再正规化模型的随机收敛？
• RQ2 来自重新居中引起的正性重整化和非高斯驱动场如何影响重整化过程及其极限？
• RQ3 对于广泛的平稳随机场，BPHZ升在分布意义上连续吗？
• RQ4 在连续性和重整化需求方面，canonical升与BPHZ升之间的关系是什么？

主要发现

• 一个普遍定理，提供在正则性结构内非线性SPDEs的再正规化模型的随机收敛结果。
• 尽管在某些Hölder空间上canonical升不连续，但BPHZ升在具有足够矩的平稳随机场上仍能在分布意义上连续。
• 该框架容纳来自非高斯驱动场和来自重新居中的正性重整化。
• 重整化常数通过指定公式计算，从而实现大部分自动化分析。
• 该方法使用类似构造性场论的多尺度分析，并针对正则性结构进行了重要的适应性改动。