Conservative stochastic PDEs on the whole space

The purpose of this paper is to establish a well-posedness theory for conservative stochastic partial differential equations on the whole space. This class of stochastic PDEs arises in fluctuating hydrodynamics, and includes the Dean--Kawasaki equation with correlated noise. In combination with the analysis of the authors and Heydecker [35], the connection between fluctuating hydrodynamics and macroscopic fluctuation theory in the context of the zero range particle process is made rigorous.