Beyond Local Solvability: Global Dynamics of Critical SPDEs via Paracontrolled Calculus

This paper explores the global dynamics of critical stochastic partial differential equations (SPDEs) by leveraging the advanced mathematical framework of paracontrolled calculus. Traditional approaches to SPDEs often face significant limitations when dealing with highly irregular noise and nonlinearities, frequently leading to results restricted to local solvability in time. Critical SPDEs, characterized by scaling invariance properties that make their solutions particularly singular, pose even greater challenges, often requiring sophisticated renormalization techniques. We present a comprehensive theoretical development of how paracontrolled calculus extends the applicability of well-posedness theory beyond these local constraints, enabling the study of long-time behavior and global properties. By systematically applying the tools of paracontrolled integration, paralinearization, and multiscale analysis, we demonstrate how to construct global solutions for a significant class of critical SPDEs, establishing their existence, uniqueness, and various qualitative features such as stability and pathwise regularity. This work aims to bridge the gap between local existence theories and the deeper understanding of global dynamical systems driven by noise, providing a robust framework for future investigations into the complex behavior of singular SPDEs.