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

1. Problem Statement and Motivation

The paper addresses the mathematical analysis of fluctuating hydrodynamics for a system of chemotactically interacting particles. The goal is to understand the continuum fluctuations of a particle system around its mean-field limit.

• The Particle System: The starting point is a system of $N$ weakly-interacting overdamped Langevin diffusions (Equation 1.2) on a $d$-dimensional torus $\mathbb{T}^d$. The particles interact via a potential derived from the Green's function of the Poisson equation, which is singular at the origin.
• The Target Equation (Dean–Kawasaki): In the limit $N \to \infty$ with noise intensity $\varepsilon = 2/N$, the system is expected to converge to the Keller–Segel–Dean–Kawasaki (KSDK) equation (Equation 1.1):
  $$(\partial_t - \Delta)\rho = -\chi\nabla \cdot (\rho \nabla \Phi_\rho) - \varepsilon^{1/2} \nabla \cdot (\sqrt{\rho}\xi)$$
  where $\xi$ is space-time white noise.
• The Analytic Challenge: The KSDK equation is notoriously difficult to analyze. It is supercritical in the sense of regularity structures (Hairer), and the multiplicative noise term $\nabla \cdot (\sqrt{\rho}\xi)$ makes the equation ill-posed unless the initial data is an empirical measure (a result by Konarovskyi, Lehmann, and von Renesse). Standard well-posedness theories fail for general initial data.
• The Proposed Solution: The authors propose an additive-noise approximation (Equation 1.4):
  $$(\partial_t - \Delta)\rho = -\chi\nabla \cdot (\rho \nabla \Phi_\rho) - \varepsilon^{1/2} \nabla \cdot (\sqrt{\rho_{\text{det}}} \xi_\delta)$$
  Here, the non-linear noise $\sqrt{\rho}\xi$ is replaced by an additive noise term driven by the square root of the deterministic Keller–Segel solution $\rho_{\text{det}}$, and the white noise $\xi$ is mollified by a correlation length $\delta$. This approximation aims to capture the correct Gaussian fluctuations while avoiding the ill-posedness of the multiplicative noise.

2. Methodology

The paper employs a dual approach, analyzing the approximation under two different regimes of relative scaling between the noise intensity $\varepsilon$ and the correlation length $\delta(\varepsilon)$.

A. The Rough Setting (Singular SPDE Theory)

• Regularity: Solutions are considered in spaces of Hölder–Besov distributions $C^{\alpha+1}(\mathbb{T}^2)$ with $\alpha+1 < -1$.
• Technique: The authors utilize the Paracontrolled Calculus framework (Gubinelli, Imkeller, Perkowski).
  • They apply the Da Prato–Debussche trick to decompose the solution into a singular stochastic part and a remainder.
  • They construct a noise enhancement (a vector of stochastic objects including iterated integrals and resonant products) that allows the equation to be solved in a distributional sense.
  • The analysis relies on the relative scaling $\varepsilon \log(\delta(\varepsilon)^{-1}) \to 0$. This scaling is "permissive" and allows for the natural scaling of particle systems ($\delta \sim N^{-1/2}$).
• Tools: Singular SPDE techniques, renormalization (though minimal due to additive noise), and the contraction principle for Large Deviation Principles (LDP).

B. The Regular Setting (Classical SPDE Theory)

• Regularity: Solutions are considered in Sobolev spaces $H^\gamma(\mathbb{T}^2)$ with $\gamma \in (-1/2, 0]$.
• Technique: Standard fixed-point arguments and energy estimates.
• Scaling: Requires a restrictive relative scaling $\varepsilon^{1/2} \delta(\varepsilon)^{-\gamma-2} \to 0$. This scaling is more stringent than the natural particle scaling but allows for higher regularity results.
• Tools: Schauder estimates, Bony's paraproduct estimates, and the weak-convergence approach to Large Deviations (Boué–Dupuis formula).

3. Key Contributions and Results

The paper establishes three fundamental probabilistic limits for the additive-noise approximation $\rho^{(\varepsilon)}_{\delta(\varepsilon)}$ as $\varepsilon \to 0$:

1. Law of Large Numbers (LLN)

• Result: The stochastic solution converges in probability to the deterministic Keller–Segel solution $\rho_{\text{det}}$.
• Spaces:
  • Rough: Convergence in $(C^{\alpha+1}(\mathbb{T}^2))^{\text{sol}}_T$ under scaling $\varepsilon \log(\delta^{-1}) \to 0$.
  • Regular: Convergence in $(H^\gamma(\mathbb{T}^2))^{\text{sol}}_T$ under scaling $\varepsilon^{1/2} \delta^{-\gamma-2} \to 0$.
• Significance: This confirms that the additive-noise approximation correctly recovers the mean-field limit.

2. Central Limit Theorem (CLT)

• Result: The rescaled fluctuations $\varepsilon^{-1/2}(\rho^{(\varepsilon)}_{\delta} - \rho_{\text{det}})$ converge to a generalized Ornstein–Uhlenbeck process $v$.
• Limit Equation: $v$ satisfies the linearized equation:
  $$(\partial_t - \Delta)v = -\chi\nabla \cdot (v \nabla \Phi_{\rho_{\text{det}}}) - \chi\nabla \cdot (\rho_{\text{det}} \nabla \Phi_v) - \nabla \cdot (\sqrt{\rho_{\text{det}}}\xi)$$
• Condition: Requires the scaling $\varepsilon^{1/2} \log(\delta^{-1}) \to 0$.
• Significance: This is a major contribution. While the CLT for the particle system (1.2) is known for repulsive interactions, it remains open for the attractive Keller–Segel case. This result provides the first rigorous derivation of the CLT for the approximation, suggesting the leading-order fluctuations of the true particle system should behave similarly.

3. Large Deviation Principle (LDP)

• Result: The family of solutions satisfies an LDP with speed $\varepsilon$ and a specific rate function $I(\rho)$.
• Rate Function: The rate function involves a "skeleton" equation driven by a Cameron–Martin element $h$:
  $$I(\rho) = \inf \left\{ \frac{1}{2}\|h\|^2_{L^2} : (\partial_t - \Delta)\rho = -\chi\nabla \cdot (\rho \nabla \Phi_\rho) - \nabla \cdot (\sqrt{\rho_{\text{det}}}h) \right\}$$
• Significance: The LDP characterizes rare events. The authors compare their rate function to the formal rate function of the original Dean–Kawasaki equation, highlighting that the approximation error lies in the skeleton equation (linear vs. fully non-linear noise term).

4. Control of Pathological Behaviors

• Negative Values: In the Regular Setting, the authors prove that the probability of the solution becoming negative (a physical impossibility for density) is exponentially small in $\varepsilon^{-1}$.
• Blow-ups: In the Rough Setting, they prove that the probability of the solution blowing up (in the distributional norm) before the deterministic blow-up time $T^*$ is also exponentially small.

4. Significance and Impact

1. Bridging Particle Systems and Continuum Models: The paper provides a rigorous mathematical bridge between the microscopic particle system (1.2) and a macroscopic continuum model that captures fluctuations. It validates the additive-noise approximation as a viable tool for studying fluctuating hydrodynamics in systems with singular interactions.
2. Handling Singular Interactions: It successfully extends the theory of singular SPDEs to the Keller–Segel context, where the interaction kernel is the Green's function of the Poisson equation (singular at the origin). This is a significant step beyond previous works that often assumed smooth interactions.
3. First-Order Accuracy: The combination of LLN and CLT results demonstrates that the additive-noise approximation is precise up to the first order of fluctuations. This suggests it can be used to study phenomena like metastability and non-equilibrium phase transitions where the Dean–Kawasaki equation is the theoretical standard but analytically intractable.
4. Methodological Advancement: The paper demonstrates the power of combining Paracontrolled Calculus (for rough paths) with classical energy methods (for regular paths) to tackle the same physical problem under different scaling regimes. It also refines the Freidlin–Wentzell theory for random variables in Wiener chaoses with specific convergence properties.

In summary, Martini and Mayorcas provide a robust theoretical framework for approximating the fluctuating hydrodynamics of chemotactic particles, overcoming the analytical barriers of the Dean–Kawasaki equation and establishing the statistical convergence properties (LLN, CLT, LDP) of their proposed model.