From Regularity to Randomness: Unifying Global Well-Posedness for Stochastic Nonlinear PDEs

This paper explores the multifaceted problem of global well-posedness for stochastic nonlinear partial differential equations (SNPDEs), aiming to unify diverse analytical frameworks across the spectrum from high regularity to inherent randomness. We address the fundamental challenges posed by the interplay of strong nonlinearities and stochastic perturbations, which often lead to solutions exhibiting low spatial or temporal regularity. The core objective is to synthesize established theories, such as those employing Itô calculus and martingale solutions, with more recent advancements like regularity structures and specialized transformations. We incorporate findings from studies on the 3D Burgers equation with multiplicative noise, where low regularity theory and a random maximum principle prove crucial, and analyses of weak turbulence models in plasma physics, which leverage connections to degenerate inhomogeneous porous medium equations for global weak solutions. By integrating these perspectives, we aim to develop a cohesive understanding of how solution properties, existence, and uniqueness are influenced by the stochastic nature of the system. The discussion highlights a theoretical framework that adapts to varying degrees of randomness and nonlinearity, contributing to a more robust and comprehensive theory of global well-posedness for SNPDEs.