正则结构的解析BPHZ定理
An analytic BPHZ theorem for regularity structures
摘要：
我们证明了一个一般性定理的随机收敛适当重整化模型所产生的非线性随机偏微分方程。正则性结构的理论给出了一个相当自动化的框架来研究这些问题，但以前的工作不得不花费大量的努力来获得这些随机估计的特设方式。相比之下，本文的主要结果就像一个黑盒子，它自动产生几乎所有符合正则结构理论范围内的方程的估计。我们的方法利用多尺度分析，强烈地让人想起在建设性域论中使用的，但有几个显着的曲折。这些特别来自于正则性结构理论的重定中心过程所引起的“正重整化”的存在，来自于可能的重整化操作组的作用的差异，以及来自于我们允许非高斯驱动场的事实。一个相当令人惊讶的事实是，尽管“正则提升”在任何包含噪声的Hölder型空间上通常都是不连续的（这就是为什么首先需要重整化），但我们证明了“BPHZ提升”，其中重整化常数是使用arXiv：1610.08468中给出的公式计算的，当限制在一类具有足够多矩的平稳随机场时，它在定律上是连续的。
英文摘要：
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.
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