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. 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 $\epsilon$ which captures the amplitude of fluctuations (tending to zero as the effective system size tends to infinity) and the correlation length $\delta$ which represents the effective scale under consideration. Let $\delta (\epsilon )\to 0$ as $\epsilon \to 0$. Under the relative scaling assumption ${\lim }_{\epsilon \to 0}\epsilon \log (\delta (\epsilon {)}^{-1})=0$ 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 ${\lim }_{\epsilon \to 0}{\epsilon }^{1\mathrm{/}2}\log (\delta (\epsilon {)}^{-1})=0$. Assuming the more restrictive relative scaling ${\lim }_{\epsilon \to 0}{\epsilon }^{1\mathrm{/}2}{\delta }^{-\gamma -2}=0$ for some $\gamma \in (-1\mathrm{/}2,0)$, 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.