Weak error analysis for a nonlinear SPDE approximation of the Dean–Kawasaki equation

We consider a nonlinear SPDE approximation of the Dean–Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order \(N^{-1-1/(d/2+1)}\log N\) $N^{-1-1/(d/2+1)}logN$ . Along the way we show well-posedness, a comparison principle, and an entropy estimate for a class of nonlinear regularized Dean–Kawasaki equations with Itô noise.