Well-Posedness of the generalised Dean-Kawasaki Equation with correlated noise on bounded domains

In this paper, we extend the notion of stochastic kinetic solutions introduced in arXiv: 2108. 08858 to establish the well-posedness of stochastic kinetic solutions of generalized Dean-Kawasaki equations with correlated noise on bounded, C²-domains with Dirichlet boundary conditions. The results apply to a wide class of non-negative boundary data, which is based on certain a priori estimates for the solutions, that encompasses all non-negative constant functions including zero and all smooth functions bounded away from zero.