Universal Central Limit Theorem for non-exchangeable interacting diffusions

Technical Summary: Universal Central Limit Theorem for Non-Exchangeable Interacting Diffusions

Problem Statement
The paper investigates the large-$n$ behavior of a system of $n$ interacting diffusions $(X^1, \dots, X^n)$ on $\mathbb{R}^d$, governed by the stochastic differential equation:
$$dX^i_t = \left( b_0(t, X^i_t) + \sum_{j=1}^n \xi_{ij} b(t, X^i_t, X^j_t) \right) dt + \sigma dB^i_t,$$
where $\xi = (\xi_{ij})$ is a deterministic interaction matrix with non-negative entries and zero diagonal. While the "Law of Large Numbers" regime (propagation of chaos) for such non-exchangeable systems has been studied extensively, particularly regarding the convergence of the empirical measure to a McKean-Vlasov limit, the behavior of the fluctuations around this limit remains less understood for heterogeneous interaction structures.

The central question addressed is whether the global fluctuation field, defined as $\eta^n_t := \sqrt{n}(\mu^n_t - \mu_t)$ (where $\mu^n_t$ is the empirical measure and $\mu_t$ the deterministic limit), converges to the same Gaussian limit as in the classical exchangeable mean-field case (where $\xi_{ij} = 1/(n-1)$). Specifically, the authors seek to determine the conditions on the interaction matrix $\xi$ under which $\eta^n$ converges in distribution to the unique solution of a linear stochastic partial differential equation (SPDE).

Methodology
The proof follows a "tightness-limit-uniqueness" strategy, conducted primarily within negative Sobolev spaces $H^{-k}$ to handle the distributional nature of the fluctuation field.

1. Tightness: The authors establish the tightness of the sequence $(\eta^n)_{n \in \mathbb{N}}$ in the space $C([0, T]; H^{-k})$. This is achieved by:

   • Proving tightness of time marginals using Fourier analysis and uniform weighted $H^1$ bounds.
   • Verifying Aldous' criterion for tightness in the path space.
   • Crucially, the authors utilize sharp quantitative propagation of chaos estimates (from Shkolnikov [58]) combined with functional inequalities (Poincaré and transport inequalities) to control the variance of the fluctuation field. They decompose the second moment of the fluctuation pairing into a variance term (controlled by Poincaré inequalities) and a bias term (controlled by the relative entropy between the particle laws and the McKean-Vlasov limit).

2. Convergence (Identification of the Limit): To identify any subsequential limit as a solution to the target SPDE, the authors analyze the semimartingale decomposition of $\langle \eta^n_t, \phi \rangle$ for test functions $\phi$.

   • The martingale part converges to a Gaussian noise term characterized by the quadratic variation.
   • The drift term involves the linearized McKean-Vlasov operator. A significant technical challenge arises because the coefficients in the drift are bounded but do not necessarily decay at infinity, while the convergence of $\eta^n$ is only in the unweighted space $H^{-k}$.
   • To resolve this, the authors employ a cutoff argument: they localize test functions on compact sets where unweighted convergence holds and control the tails using uniform weighted moment estimates derived from the Bessel kernel properties and transport inequalities.
   • They show that the error terms arising from the heterogeneity of $\xi$ vanish in the limit under the specified denseness conditions.

3. Uniqueness: The paper proves pathwise uniqueness for the limiting SPDE. By considering the difference $\delta_t = \eta_t - \eta'_t$ between two solutions driven by the same noise, the noise terms cancel, leaving a deterministic linear equation. The authors derive an energy inequality in a weighted negative Sobolev norm and apply Gronwall's inequality to show that $\delta_t \equiv 0$.

Key Assumptions and Conditions
The main result holds for interaction matrices $\xi$ satisfying Assumption 2.1, which includes:

• Row Sums: $\sum_j \xi_{ij} = 1$ for all $i$.
• Column Sums: Bounded column sums, with the average squared column sum converging to 1.
• Denseness Threshold: The matrix entries must satisfy a specific decay condition. The most critical condition is that the maximum entry of $\xi$ must be $o(n^{-1/2})$. Specifically, the condition $\frac{1}{n^{3/2}} (\sum_{i,j} |\tilde{\xi}_{ij}|) \max_{i,j} \xi_{ij} \to 0$ is required, where $\tilde{\xi}$ captures the deviation from the mean-field case.
• Regularity: The interaction functions $b_0$ and $b$ must have bounded derivatives up to order $k+1$.

Main Results
The paper establishes a Universal Central Limit Theorem (Theorem 2.4):

• Under the stated structural and denseness conditions on $\xi$, the fluctuation field $\eta^n$ converges in distribution in $C([0, T]; H^{-k})$ to a unique Gaussian process $\eta$.
• This limit $\eta$ is the unique solution to the linear SPDE:
  $$\partial_t \eta_t = \frac{\sigma^2}{2} \Delta \eta_t - \nabla \cdot (b_0 \eta_t) - \nabla \cdot \left( \eta_t \int b(\cdot, y) \mu_t(dy) \right) - \nabla \cdot \left( \mu_t \int b(\cdot, y) \eta_t(dy) \right) - \sigma \nabla \cdot (\sqrt{\mu_t} \dot{W}),$$
  which is identical to the fluctuation limit in the exchangeable mean-field case.
• The result applies to:
  • Scaled adjacency matrices of $m_n$-regular graphs where $m_n / \sqrt{n} \to \infty$.
  • Erdős–Rényi graphs $G(n, p_n)$ where $n p_n / \sqrt{n} \to \infty$ (in the quenched sense).
  • Spatial interaction models in the subcritical regime ($\alpha < 1/2$).

Significance and Sharpness
The paper claims that the $n^{-1/2}$ threshold for the maximum entry of the interaction matrix is sharp.

• In the context of spatial interaction models (where interaction strength decays as $|x_i - x_j|^{-\alpha}$), the condition $\max \xi_{ij} \sim n^{\alpha-1}$ implies that the $n^{-1/2}$ threshold corresponds to $\alpha < 1/2$.
• The authors note that in the supercritical regime ($\alpha > 1/2$), the fluctuations, when rescaled by $\sqrt{n}$, diverge, and a different scaling ($n^{1-\alpha}$) leads to a deterministic limit rather than a Gaussian one.
• The universality of the Gaussian limit is significant because it demonstrates that, provided the interaction graph is sufficiently dense (specifically, the interaction strength is not too concentrated on any single pair), the microscopic heterogeneity of the network does not alter the macroscopic fluctuation statistics; the system behaves as if it were fully exchangeable.

Conclusion
This work extends the classical fluctuation theory of mean-field interacting diffusions to a broad class of non-exchangeable systems. By combining quantitative propagation of chaos with functional inequalities and a careful analysis in weighted Sobolev spaces, the authors prove that the universality of the Gaussian fluctuation limit holds as long as the interaction matrix satisfies a specific denseness condition, which is shown to be optimal for spatial models.