Mean field limit for disordered diffusions with singular interactions

Abstract

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in Rm in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice Zd , 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 < d. Our framework cover the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators. © Institute of Mathematical Statistics, 2014.