Renormalization of Singular Stochastic PDEs and the Theory of Paracontrolled Distributions

This monograph presents a rigorous analysis of singular stochastic partial differential equations (SPDEs) within the framework of paracontrolled distributions. We address the fundamental ill-posedness inherent in non-linear equations driven by rough noise, such as the Kardar-Parisi-Zhang (KPZ) equation and the dynamical 43 model. Classical analytical techniques fail due to the proliferation of ultraviolet divergences arising from the product of distributions. We systematically develop the theory of paracontrolled calculus, leveraging Bony's paradifferential operators and Littlewood-Paley decomposition to disentangle the singular structure of solutions. A central focus is the renormalization procedure, necessitating the introduction of infinite counterterms to restore well-posedness. We establish the existence and uniqueness of solutions to renormalized equations in appropriate Besov spaces, providing a lightweight yet powerful alternative to the theory of regularity structures. The exposition culminates in the convergence of smooth approximations to the canonical stochastic objects.