Mean field limit for disordered diffusions with singular interactions

TL;DR

This paper proves the convergence of large populations of spatially extended, disordered diffusions with singular interactions to a deterministic McKean-Vlasov equation, including cases with degenerate noise and power-law interactions relevant to neural models.

Contribution

It establishes the mean field limit for singular, spatially extended diffusions in disordered environments, including convergence rates and fluctuation analysis.

Findings

Convergence of empirical measures to McKean-Vlasov solutions.

Well-posedness of the McKean-Vlasov equation even without noise.

Quantitative estimates of convergence speed and fluctuation behavior.

Abstract

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf {R}^m$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf {Z}^d$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<\alpha<d$), we address the convergence as $N\to\infty$ of the empirical measure of the diffusions to the solution of a deterministic McKean-Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when $\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.