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 ε 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(ε) → 0 as ε → 0. Under the relative scaling assumption lim_ε→0 ε log(δ(ε)⁻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_ε→0 ε^1/2 log(δ(ε)⁻¹ ) = 0. Assuming the more restrictive relative scaling lim_ε→0 ε^1/2 δ^−γ−2 = 0 for some γ ∈ (−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.