During the review period, the project achieved substantial progress in the analysis of singular conservative Stochastic Partial Differential Equations (SPDEs). The work produced new results on well-posedness, regularity, large deviations, fluctuation behavior, and stochastic dynamics across several major model classes.
For stochastic thin-film equations, which are highly degenerate as the fluid height approaches vacuum, we established existence of weak solutions for a broad family of models with multiplicative noise and nonclassical mobility. The theory accommodates irregular initial data, including vanishing profiles and mass concentrations. A new compactness framework based on entropy bounds and mass conservation enabled these results and shows that required noise regularity can be significantly relaxed.
In the analysis of stochastic gradient descent (SGD), we developed a rigorous scaling limit showing that SGD evolves as a nonlinear SPDE capturing both macroscopic drift and intrinsic fluctuations. We proved optimal-rate convergence of the empirical parameter distribution and identified second-order effects absent from deterministic mean-field limits. A coupling-based well-posedness argument allows treatment of irregular coefficients, contributing to a unified theoretical description of SGD in overparameterized regimes.
For conservative SPDEs on unbounded domains, including generalized Dean-Kawasaki models on ℝᵈ, the project produced a comprehensive solution theory. By extending kinetic formulations and employing renormalization and spatial cutoffs, the analysis covers both low- and high-density regimes. A pathwise L¹-contraction principle yields strong uniqueness. We also established a large deviation principle linking microscopic particle models with macroscopic fluctuation theory.
In regularity theory, we obtained optimal space-time regularity for generalized porous-medium-type equations with nonlocal and spatially heterogeneous structure. The proofs rely on refined microlocal techniques and a new parametrix construction that avoids earlier technical assumptions. In parallel, we introduced a real-analysis method for treating operators with extremely irregular coefficients without Fourier tools, using a structural factorization of the equations.
The project further advanced the understanding of supercritical limits and fluctuation expansions. For the Landau-Lifshitz–Navier–Stokes system, we proved a dynamic large deviation principle under joint vanishing-noise and vanishing-correlation scaling and uncovered a structural relation linking large deviations, weak–strong uniqueness, and deterministic energy equalities.
Finally, in stochastic dynamics, we completed the first quantitative study of noise-induced synchronization in vector-valued stochastic reaction-diffusion systems. For potentials with degenerate minima, explicit bounds on the top Lyapunov exponent show that synchronization is driven by curvature of the minima manifold. Additional results include a low-temperature asymptotic expansion for the Euclidean Φ⁴2 measure-yielding a Law of Large Numbers and a Central Limit Theorem-and the development of a random dynamical systems framework for McKean-Vlasov SDEs, establishing a perfect cocycle via pathwise rough-path methods.
