Large deviations for cascades of diffusions arising in oscillating systems of interacting Hawkes processes

TL;DR

This paper investigates the large deviations of diffusion approximations in oscillatory systems of interacting Hawkes processes, revealing the probabilistic behavior of large populations in neural models with memory effects.

Contribution

It provides the first large deviation principle for the invariant measure of hypo-elliptic diffusions arising from oscillating Hawkes process systems.

Findings

Established a large deviation principle for the invariant measure.

Analyzed the hypo-elliptic diffusion with memory terms.

Connected the results to neural network modeling.

Abstract

We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Loecherbach (2017) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high dimensional chain of differential equations driven by $2-$dimensional Brownian motion. We study the large-population-, i.e., small noise-limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.