Regularity Structures for Quasilinear Singular Stochastic PDEs and the Generalized KPZ Equation

We develop a transfinite extension of the theory of regu-larity structures to address a class of quasilinear singularstochastic partial differential equations (SPDEs) in thesubcritical regime. Specifically, we focus on the general-ized Kardar-Parisi-Zhang (gKPZ) equation with solution-dependent diffusion coefficients. The presence of the quasi-linear term breaks the translation invariance of the renor-malization group flow, necessitating a local renormaliza-tion procedure that depends on the solution manifold.We construct a family of solution-dependent models Muand establish a fixed-point theorem in a space of modeleddistributions Dγ,η equipped with a solution-dependenttopology. We prove local existence and uniqueness of therenormalized solutions and provide sharp Schauder-typeestimates for the associated reconstruction operator R.