Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general $C^2$ bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.