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The results of the author and Gess [27] develop a robust well-posednesstheory for a broad class of conservative stochastic PDEs, with bothprobabilistically stationary and non-stationary Stratonovich noise, and withirregular noise coefficients like the square root. However, one case leftuntreated by [27] is the case of SPDEs that combine conservative,non-stationary Stratonovich noise with square root-like nonlinearities. Suchequations arise naturally in the fluctuating hydrodynamics of inhomogenoussystems, and a new analysis is required to handle certain discontinuouscoefficients appearing in their It\^o formulations. We treat thediscontinuities by showing that the equation exhibits a novel regularization ofthe logarithm of the solution, and establish the well-posedness by building onthe concept of a stochastic kinetic solution introduced in [27].

Stochastic PDEs with correlated, non-stationary Stratonovich noise of Dean--Kawasaki type