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 $\varepsilon$ 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.

💡 Research Summary

This paper tackles the analytical challenges posed by the Keller‑Segel‑Dean‑Kawasaki (KS‑DK) stochastic partial differential equation, whose nonlinear chemotactic drift and multiplicative Dean‑Kawasaki noise make the system super‑critical and ill‑posed in classical function spaces. To circumvent these difficulties, the authors introduce an additive‑noise approximation: the stochastic term is linearized by replacing the original noise ∇·(√ρ ξ) with ∇·(√ρ_det ξ_δ), where ρ_det is the deterministic Keller‑Segel solution and ξ_δ is a spatially mollified white noise at scale δ. Two small parameters govern the model: the noise intensity ε and the correlation length δ, with δ(ε)→0 as ε→0.

The core of the analysis lies in three relative scaling regimes:

1. Law of Large Numbers (LLN) regime – ε log δ⁻¹ → 0. Under this condition the “noise enhancement” term vanishes in probability, allowing the authors to employ the paracontrolled calculus (Gubinelli‑Imkeller‑Perkowski) to prove that the solution map is locally Lipschitz in a space of Hölder‑Besov distributions C^{α+1} (α∈(‑9/4,‑2)). Consequently, the stochastic solution ρ^{ε}_δ converges in probability to the deterministic Keller‑Segel solution ρ_det for any fixed time horizon T strictly before the possible blow‑up time T* of ρ_det.

2. Central Limit Theorem (CLT) regime – ε^{1/2} log δ⁻¹ → 0. Here the authors rescale the fluctuation v^{ε}=ε^{-1/2}(ρ^{ε}_δ‑ρ_det) and show that v^{ε} converges in law to a generalized Ornstein‑Uhlenbeck process solving a linear SPDE with coefficients depending on ρ_det. This result matches the Gaussian fluctuations predicted by fluctuating hydrodynamics and extends recent CLT results for repulsive chemotaxis to the attractive case, at least at the level of the additive approximation.

3. Higher‑regularity regime – ε^{1/2} δ^{-γ‑2} → 0 for some γ∈(‑½,0). This stronger scaling permits the analysis in Sobolev spaces H^{γ}, yielding LLN, CLT, and Large Deviation Principle (LDP) results in regular function spaces. The authors control the negative part of the solution in L² and prove that the probability of a blow‑up in the Besov norm before T is exponentially small in ε^{-1}. They also establish an LDP with speed ε and rate function
   I(ρ)=inf_{h∈L²}½‖h‖²,
   where h appears in the deterministic skeleton equation (∂_t‑Δ)ρ = –χ∇·(ρ∇Φ_ρ) – ∇·(√ρ_det h). Compared with the fully nonlinear skeleton arising from the original Dean‑Kawasaki noise, the additive model’s skeleton is linear in the Cameron‑Martin direction, highlighting the approximation error.

Methodologically, the paper blends singular SPDE techniques (regularity structures/paracontrolled distributions) with probabilistic tools (Girsanov transform, contraction principle, sub‑Gaussian tail estimates). It carefully treats the singular Green’s function of the Poisson equation, the blow‑up of the deterministic Keller‑Segel solution in two dimensions, and the necessity of working on time intervals