Stochastic PDEs with correlated, non-stationary Stratonovich noise of Dean--Kawasaki type
Benjamin Fehrman
TL;DR
The article advances the well-posedness theory for conservative SPDEs driven by correlated, non-stationary Stratonovich noise of Dean–Kawasaki type. It develops and employs a stochastic kinetic solution framework to handle square-root nonlinearities and discontinuous coefficients arising in the Itô form, establishing existence, uniqueness, and pathwise stability while revealing a log-regularization mechanism and time-averaged parabolic regularity. The work extends prior results (FG21) to Neumann boundary conditions and non-stationary noise, and demonstrates a robust approach to inhomogeneous diffusion, nonnegativity preservation, and a dichotomy for the zero-set of solutions. These results have implications for fluctuating hydrodynamics and macroscopic fluctuation theory in inhomogeneous media, providing a rigorous basis for density fluctuation models with spatially varying diffusivity and correlated noise.
Abstract
The results of the author and Gess [27] develop a robust well-posedness theory for a broad class of conservative stochastic PDEs, with both probabilistically stationary and non-stationary Stratonovich noise, and with irregular noise coefficients like the square root. However, one case left untreated by [27] is the case of SPDEs that combine conservative, non-stationary Stratonovich noise with square root-like nonlinearities. Such equations arise naturally in the fluctuating hydrodynamics of inhomogenous systems, and a new analysis is required to handle certain discontinuous coefficients appearing in their Itô formulations. We treat the discontinuities by showing that the equation exhibits a novel regularization of the logarithm of the solution, and establish the well-posedness by building on the concept of a stochastic kinetic solution introduced in [27].