Quenched asymptotics for interacting diffusions on inhomogeneous random graphs

TL;DR

This paper studies the large-population behavior of interacting diffusions on inhomogeneous random graphs, proving quenched convergence of empirical measures to a nonlinear PDE, with applications in neuroscience.

Contribution

It extends previous homogeneous graph results to inhomogeneous, disordered graphs, establishing quenched convergence to nonlinear PDEs including neural field equations.

Findings

Empirical measures converge to a nonlinear Fokker-Planck PDE.

The spatial profile converges to a nonlinear integro-differential equation.

Results apply to disordered W-random graphs with unbounded graphons.

Abstract

The aim of the paper is to address the behavior in large population of diffusions interacting on a random, possibly diluted and inhomogeneous graph. This is the natural continuation of a previous work, where the homogeneous Erd\H os-R\'enyi case was considered. The class of graphs we consider includes disordered $W$-random graphs, with possibly unbounded graphons. The main result concerns a quenched convergence (that is true for almost every realization of the random graph) of the empirical measure of the system towards the solution of a nonlinear Fokker-Planck PDE with spatial extension, also appearing in different contexts, especially in neuroscience. The convergence of the spatial profile associated to the diffusions is also considered, and one proves that the limit is described in terms of a nonlinear integro-differential equation which matches the neural field equation in certain particular cases.