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

TL;DR

This paper analyzes an additive-noise approximation to Keller-Segel-Dean-Kawasaki dynamics, deriving limit theorems and large deviation principles for small noise and correlation length scales, with implications for particle system fluctuations.

Contribution

It introduces a novel approximation framework for chemotactic particle dynamics and establishes rigorous probabilistic limit results under various scaling regimes.

Findings

Law of large numbers and large deviation principles derived in irregular distribution spaces.

Central limit theorem established under specific scaling assumptions.

Results inform continuum fluctuation analysis of particle systems.

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 r\^{o}le 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 $\delta$ which represents the effective scale under consideration. Let $\delta(\varepsilon)\to0$ as $\varepsilon\to0$. Under the relative scaling assumption $\lim_{\varepsilon\to0}\varepsilon\log(\delta(\varepsilon)^{-1})=0$ 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 $\lim_{\varepsilon\to0}\varepsilon^{1/2}\log(\delta(\varepsilon)^{-1})=0$. Assuming the more restrictive relative scaling $\lim_{\varepsilon\to0}\varepsilon^{1/2}\delta^{-\gamma-2}=0$ for some $\gamma\in(-1/2,0)$, 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.