Malliavin Calculus and Regularity Structures for Quasilinear Stochastic PDEs with Rough Noise

We develop a robust framework for the analysis of quasilinear stochastic partial differential equations (SPDEs) driven by rough noise, specifically focusing on the interplay between Malliavin calculus and the theory of regularity structures. We consider equations of the form t u - a(u) u = , where denotes space-time white noise or a rough fractional Gaussian field, and the diffusion coefficient a(u) introduces a quasilinear dependence that destroys the Gaussian structure of the solution manifold. By lifting the renormalization group flow to the tangent bundle of the model space, we construct a Malliavin derivative operator compatible with the BPHZ renormalization procedure. We establish the existence of a renormalized solution map that is Malliavin differentiable and prove a Hörmander-type theorem, ensuring the absolute continuity of the law of the solution with respect to the Lebesgue measure. Our results bridge the gap between the analytic constraints of singular SPDEs and the probabilistic tools required for density estimates in non-Gaussian settings.