The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.

中文翻译：

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正则化结构中的 SPDE 重整化

最近在 arXiv:1610.08468 中引入的形式主义允许为半线性随机偏微分方程的任何亚临界系统分配一个正则结构以及相应的“重整化群”。在非常温和的附加假设下，然后在 arXiv:1612.08138 中表明，在重新归一化程序之后，表现出相关小规模行为的大类驾驶噪声可以以稳健的方式提升到这样的规律性结构，这让人联想到 BPHZ 程序出现在微扰 QFT 中。目前的工作通过将重整化群的动作构造到合适的随机偏微分方程类上来完成该程序，该随机偏微分方程与其在相应模型空间上的动作交织在一起。这特别表明，由平滑驱动噪声的 BPHZ 提升构造的解与修改的 PDE 的经典解一致。这为一大类奇异非线性 SPDE 产生了一个非常通用的黑盒类型局部存在和稳定性定理。