Hairer’s regularity structures provide a general framework for a solution theory for partial differential equations driven by a noise that is so rough that the nonlinearity requires a renormalization. The notions (like the structure group) and tools (like modelled distributions) are surprisingly versatile and not tied to the combinatorics of decorated trees and diagrams, but extend to more top-down ways of organizing renormalization. In fact, they are in harmony with a more analytic approach, based on taking derivatives with respect to the constitutive law defining the nonlinearity and with respect to the noise (Malliavin derivative).
