Rigorous derivation of the mean-field limit for the signal-dependent Keller-Segel system

Technical Summary: Rigorous Derivation of the Mean-Field Limit for the Signal-Dependent Keller-Segel System

Problem Statement
This work addresses the rigorous derivation of a two-dimensional Keller–Segel system with signal-dependent sensitivity from a stochastic interacting particle model. The target macroscopic system is given by:
$$\begin{cases} \partial_t u = \Delta(e^{-v}u + u), \\ -\Delta v + v = \chi u, \end{cases}$$
where $u$ represents cell density and $v$ the chemical signal concentration. The model describes biological phenomena such as stripe pattern formation via self-trapping mechanisms (e.g., E. coli secreting AHL), where bacterial motility decreases as signal concentration increases.

The microscopic system consists of $N$ interacting particles $\{X^{\varepsilon}_{N,i}\}_{1\le i \le N}$ in $\mathbb{R}^2$ governed by stochastic differential equations (SDEs) with diffusion coefficients dependent on the empirical mean-field interaction:
$$dX^{\varepsilon}_{N,i}(t) = \left(2 \exp\left(-\frac{1}{N}\sum_{j=1}^N \Phi^{\varepsilon}(X^{\varepsilon}_{N,i} - X^{\varepsilon}_{N,j})\right) + 2\right)^{1/2} dB_i(t),$$
where $\Phi^{\varepsilon}$ is a regularized potential. The central challenge is to prove that as $N \to \infty$, the particle system converges to the mean-field limit (a non-linear SDE system) and subsequently to the PDE solution, specifically under an algebraic scaling regime for the regularization parameter $\varepsilon$ (i.e., $\varepsilon \sim N^{-\gamma}$), improving upon previous results that required logarithmic scaling.

Methodology
The authors employ a multi-step analytical framework combining probabilistic estimates, stopping time techniques, and relative entropy methods:

1. Intermediate Mean-Field System: The authors introduce an intermediate non-local PDE system (1.9) and its associated non-linear SDE (1.8). This system serves as a bridge between the $N$-particle system and the limiting PDE (1.1). The existence and uniqueness of weak solutions for these intermediate systems are established based on prior work (specifically [2]), utilizing bounds on $\nabla \log u^{\varepsilon}$.

2. Convergence in Probability (Section 2):

   • Stopping Times: To handle the non-linear dependence of diffusion coefficients on particle interactions, the authors define a stopping time $\tau_{\alpha}$ that halts the process if the maximum distance between particle trajectories and mean-field trajectories exceeds $N^{-\alpha}$. This technique allows for the application of the Law of Large Numbers (LLN) within a controlled regime.
   • Burkholder-Davis-Gundy (BDG) Inequality: Due to the non-linear diffusion, the BDG inequality is essential for estimating the stochastic integrals.
   • Error Decomposition: The difference between the particle system and the mean-field limit is decomposed into terms involving the empirical measure and the convolution with the density. The authors utilize Lemma 2 (LLN estimates from [12]) to bound the probability of large deviations.
   • Scaling Analysis: By carefully selecting parameters $\theta, \alpha, k, m$, and the scaling exponent $\gamma$, the authors derive algebraic convergence rates, ensuring that error terms decay as negative powers of $N$.

3. Strong $L^1$ Propagation of Chaos (Section 3):

   • Relative Entropy Method: Building on the convergence in probability, the authors apply the relative entropy method (refined in [11]) to establish strong convergence. They define the relative entropy $H(u^{\varepsilon}_N | u^{\varepsilon \otimes N})$ between the joint density of the particle system and the tensor product of the mean-field density.
   • Entropy Dissipation: By differentiating the relative entropy with respect to time and applying integration by parts, the authors derive a differential inequality. The negative terms (dissipation) are controlled by the Fisher information, while the error terms are bounded using the probabilistic convergence results from Section 2 and the properties of the regularized potential $\Phi^{\varepsilon}$.
   • Csiszár-Kullback-Pinsker (C-K-P) Inequality: This inequality is used to translate the relative entropy bound into an $L^1$ norm bound for the marginal densities, quantifying the propagation of chaos.

Key Contributions and Results

• Algebraic Scaling Improvement: The primary contribution is the derivation of the mean-field limit under an algebraic scaling $\varepsilon \sim N^{-\gamma}$ (where $\gamma > 0$). This improves upon existing results (e.g., [2]) which were limited to logarithmic scaling ($\varepsilon \sim (\log N)^{-1/4}$).
• Convergence in Probability (Theorem 1): The paper proves that for specific parameter ranges, the probability that the maximum deviation between the $N$-particle system and the mean-field SDE exceeds $N^{-\alpha}$ decays algebraically:
  $$P\left(\max_{i} |X^{\varepsilon}_{N,i} - X^{\varepsilon}_i| > N^{-\alpha}\right) \le C N^{-\eta}.$$
• Strong $L^1$ Propagation of Chaos (Theorem 2): The authors establish a quantitative strong convergence rate for the propagation of chaos. They show that the $L^1$ distance between the $r$-th marginal of the particle system and the tensor product of the mean-field solution satisfies:
  $$\|u^{\varepsilon}_{N,r} - u^{\varepsilon \otimes r}\|^2_{L^\infty(0, T^*; L^1)} \le C \varepsilon^{\beta},$$
  where $\beta$ is a positive exponent determined by the scaling parameters. In the specific numerical example provided, this yields a rate of $O(\varepsilon^{46})$, significantly faster than the $O(\varepsilon)$ rate found in previous literature under logarithmic scaling.
• Parameter Regime: The paper provides explicit conditions on the parameters $\theta, \alpha, k, m, \gamma,$ and $\eta$ required to achieve these convergence rates, demonstrating the feasibility of the algebraic scaling.

Significance
The paper claims to provide a more refined and rigorous derivation of the signal-dependent Keller-Segel system from microscopic stochastic dynamics. By introducing stopping times and leveraging the Law of Large Numbers effectively, the authors overcome the limitations of logarithmic scaling found in prior works. The application of the relative entropy method to this specific signal-dependent context allows for the establishment of strong $L^1$ convergence with explicit algebraic rates. This work bridges the gap between microscopic particle interactions and macroscopic PDE descriptions in a regime that is mathematically more robust and potentially more applicable to physical systems where algebraic scaling is preferred over logarithmic. The results rely on the well-posedness of the intermediate PDEs established in [2] and extend the methodology of moderately interacting particle systems to handle signal-dependent sensitivities with improved convergence guarantees.