Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

Abstract

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-H\"older 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.