Well-Posedness of the Dean–Kawasaki and the Nonlinear Dawson–Watanabe Equation with Correlated Noise

In this paper we prove the well-posedness of the generalized Dean–Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally \({1}/{2}\) $1/2$ -Hölder continuous, including the square root. This solves several open problems, including the well-posedness of the Dean–Kawasaki equation and the nonlinear Dawson–Watanabe equation with correlated noise.