Propagation of Chaos for Stochastic Spatially Structured Neuronal Networks with Delay driven by Jump Diffusions

TL;DR

This paper proves the well-posedness and propagation of chaos for a class of stochastic neural networks with spatial structure, delay, and jump diffusion noise, introducing a novel Euler approximation method for McKean-Vlasov equations.

Contribution

It establishes the first Euler approximation-based existence proof for McKean-Vlasov equations with delay and jump diffusion, and demonstrates propagation of chaos in this complex neural network model.

Findings

Well-posedness of the McKean-Vlasov equation with delay and jump noise

Propagation of chaos in infinite neural networks with spatial structure

Introduction of Euler approximation for such equations

Abstract

Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean-Vlasov equation and a corresponding propagation of chaos result in the infinite population limit are proven. Our existence result for the McKean-Vlasov equation is based on the Euler approximation, that is applied to this type of equation for the first time.