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

Abstract

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. As such, the interaction potential is given by the Green's function associated to Poisson's equation, which is singular around the origin. Two parameters play a key rôle in the approximation: the noise intensity $\varepsilon$ which captures the amplitude of fluctuations (tending to zero as the effective system size tends to infinity) and the correlation length $δ$ which represents the effective scale under consideration. Let $δ(\varepsilon)\to0$ as $\varepsilon\to0$. Under the relative scaling assumption $\lim_{\varepsilon\to0}\varepsilon\log(δ(\varepsilon)^{-1})=0$ we obtain analogues of law of large numbers and large deviation principles in irregular spaces of distributions using methods of singular stochastic partial differential equations. The same techniques also yield a central limit theorem under the relative scaling $\lim_{\varepsilon\to0}\varepsilon^{1/2}\log(δ(\varepsilon)^{-1})=0$. Assuming the more restrictive relative scaling $\lim_{\varepsilon\to0}\varepsilon^{1/2}δ^{-γ-2}=0$ for some $γ\in(-1/2,0)$, we also obtain analogues of law of large numbers and large deviation principles in regular function spaces using a mixture of pathwise and probabilistic tools. We further describe consequences of these results relevant to applications of our approximation in studying continuum fluctuations of particle systems.