An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions

1. Problem Statement and Motivation

The paper addresses the mathematical challenge of defining solutions to a singular stochastic partial differential equation (SPDE) that models the fluctuating hydrodynamics of the Keller–Segel model for chemotaxis in two dimensions.

• The Model: The core equation is a parabolic-elliptic system on the 2D torus $\mathbb{T}^2$:
  $$\begin{cases} (\partial_t - \Delta)\rho = \nabla \cdot (\rho \nabla \Phi_\rho) + \nabla \cdot (\sigma \xi), \\ -\Delta \Phi_\rho = \rho - \langle \rho, 1 \rangle_{L^2}, \\ \rho|_{t=0} = \rho_0, \end{cases}$$
  where $\xi$ is space-time white noise, $\sigma$ is a space-time heterogeneity (ideally $\sigma = \sqrt{\rho}$ for the Dean–Kawasaki equation), and $\Phi_\rho$ is the potential solving the elliptic equation.
• The Difficulty: The equation is singular. The noise $\xi$ has regularity $C^{-2-}$, implying the solution $\rho$ has regularity roughly $C^{-1-}$. The non-linear drift term $\rho \nabla \Phi_\rho$ involves the product of a distribution ($\rho$) and a function with negative regularity ($\nabla \Phi_\rho \in C^{0-}$). By standard distribution theory (Bony's estimates), this product is ill-defined.
• The Goal: The authors aim to establish local well-posedness for an additive-noise approximation of this system. Instead of the multiplicative noise $\nabla \cdot (\sqrt{\rho}\xi)$, they consider $\nabla \cdot (\sigma \xi)$ where $\sigma$ is a fixed, sufficiently regular function (specifically $\sigma = \sqrt{\rho_{\text{det}}}$, the square root of the deterministic solution). This approximation is crucial for proving limit theorems (Law of Large Numbers, Central Limit Theorem, Large Deviation Principle) in a follow-up work.

2. Methodology: Paracontrolled Distributions

The authors employ the theory of Paracontrolled Distributions (Gubinelli, Imkeller, Perkowski), a framework designed to handle singular SPDEs by decomposing solutions into a "singular" part driven by the noise and a "regular" remainder.

• The Ansatz: The solution $\rho$ is decomposed as:
  $$\rho = \mathcal{X} + \mathcal{Y} + w$$
  where $\mathcal{X}$ and $\mathcal{Y}$ are specific stochastic objects derived from the noise (the "enhancement"), and $w$ is a remainder term with higher regularity. The term $w$ is further decomposed using a Gubinelli derivative $w'$ to handle the resonant products.
• Renormalization: Due to the singularity, the equation must be renormalized. The authors introduce a sequence of counterterms (diverging fields) $f_\delta$ to subtract the infinite parts arising from the product of the noise and the solution.
• Key Innovation (Symmetry): A major technical hurdle in singular SPDEs is the divergence rate of renormalization constants. Standard power counting suggests a linear divergence ($\delta^{-1}$) for the relevant terms. However, the authors exploit the symmetry of the elliptic Green's function (Poisson equation) to show that these divergences cancel out partially.
  • Result: The divergence is reduced from linear ($\delta^{-1}$) to logarithmic ($\log(\delta^{-1})$).
  • Special Case: If the heterogeneity $\sigma$ is constant, the symmetry leads to perfect cancellation, and no renormalization counterterm is needed at all.

3. Key Contributions

1. Local Well-Posedness: The paper proves the existence and uniqueness of a paracontrolled solution to the singular SPDE (1.1) on a random time interval $[0, T_{\max})$. The solution is constructed as a limit of mollified equations.
2. Logarithmic Renormalization: The authors establish that the renormalization field $f_\delta$ diverges at most logarithmically with the mollification parameter $\delta$. This is a significant improvement over the expected linear divergence and is a direct consequence of the specific structure of the Keller–Segel non-linearity combined with the elliptic Green's function symmetry.
3. Handling Heterogeneity: Unlike many previous works that assume constant noise coefficients, this paper handles heterogeneous noise ($\sigma \in C([0,T]; H^2(\mathbb{T}^2))$). This is essential for the application to the Dean–Kawasaki equation where $\sigma$ depends on the density.
4. Noise Enhancement Construction: The paper provides a rigorous construction of the enhanced noise (the collection of stochastic objects required to define the solution), including:
   • $\mathcal{X} = \nabla \cdot I[\sigma \xi]$
   • $\mathcal{Y} = \nabla \cdot I[\mathcal{X} \nabla \Phi \mathcal{X}] - \mathbb{E}[\dots]$
   • Resonant products and commutators involving these terms.
     They prove these objects exist in appropriate Besov spaces and converge as the mollification vanishes.

4. Main Results

• Theorem 1.2 (Main Result):

  • Existence & Uniqueness: For suitable initial data $\rho_0$ and heterogeneity $\sigma$, there exists a unique paracontrolled solution $\rho$ to the renormalized equation.
  • Maximal Time: The solution exists up to a maximal time $T_{\max}$, which is either the full time horizon $T$ or the time when the solution blows up in the $C^{\alpha+1}$ norm.
  • Convergence: The solutions $\rho_\delta$ to the mollified equations converge in probability to the renormalized solution $\rho$ as $\delta \to 0$.
  • Renormalization Bound: The diverging counterterm $f_\delta$ satisfies $\|f_\delta\| \lesssim (1 \vee \log(\delta^{-1})) \|\sigma\|^2$.

• Technical Estimates:

  • The authors derive precise bounds for "shape coefficients" (integrals arising from Feynman diagrams) that capture the time and frequency dependencies of the stochastic terms.
  • They utilize Feynman diagram techniques to organize the stochastic calculus, identifying which terms require renormalization and which cancel out due to symmetry.

5. Significance and Impact

• Bridging Micro and Macro: This work provides the rigorous analytical foundation for approximating the Dean–Kawasaki equation (which describes fluctuations of interacting particle systems) using additive noise. This is a critical step in connecting microscopic particle systems to macroscopic fluctuating hydrodynamics.
• Anisotropic Regularity: The paper highlights the utility of the paracontrolled approach for equations with anisotropic regularity (where time and space regularize differently), which is characteristic of parabolic-elliptic systems. The authors argue that this approach is more natural for such systems than the theory of Regularity Structures.
• Future Applications: The results are directly applied in a follow-up paper ([MM24]) to prove:
  • A Law of Large Numbers (convergence to the deterministic Keller–Segel equation).
  • A Central Limit Theorem (fluctuations described by a generalized Ornstein–Uhlenbeck process).
  • A Large Deviation Principle.
• Methodological Advancement: The demonstration that elliptic symmetries can reduce divergence orders from linear to logarithmic offers a new perspective on renormalization in singular SPDEs, potentially applicable to other models involving non-local operators.

In summary, this paper successfully tames a highly singular, non-local, and heterogeneous SPDE arising in biological modeling, providing a robust framework for analyzing fluctuations in chemotaxis systems.