An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Small-Noise Results

We study an additive-noise approximation to Keller--Segel--Dean--Kawasaki dynamics which is proposed as an approximate model to the fluctuating hydrodynamics of chemotactically interacting particles around their mean-field limit. Two parameters play a key r\^ole in the approximation: the noise intensity which captures the amplitude of these fluctuations (tending to zero as the effective system size tends to infinity) and the correlation length which represents the effective scale under consideration. Using a mixture of pathwise and probabilistic tools we obtain analogues of law of large numbers and large deviation principles in regular function spaces under restrictive relative scaling assumptions on the noise intensity and the correlation length. Using methods of singular stochastic partial differential equations we also obtain analogues of law of large numbers, central limit theorems and large deviation principles in irregular spaces of distributions under general relative scaling assumptions. We further describe consequences of these results relevant to applications of our approximation in studying continuum fluctuations of particle systems.