Mean-field limits for non-linear Hawkes processes with excitation and inhibition

TL;DR

This paper analyzes the mean-field limits of multivariate non-linear Hawkes processes with mixed excitation and inhibition, revealing different behaviors in subcritical and critical cases, including deterministic and stochastic limit equations.

Contribution

It introduces a novel analysis of the mean-field limit for Hawkes processes with mixed excitation and inhibition, distinguishing between deterministic and stochastic limits based on the excitation-inhibition balance.

Findings

In the subcritical case ($p eq 1/2$), the limit is deterministic with independent components.

In the critical case ($p=1/2$), the limit is stochastic with conditionally independent components.

The limit equations are convolution equations, either deterministic or stochastic.

Abstract

We study a multivariate, non-linear Hawkes process $Z^N$ on the complete graph with $N$ nodes. Each vertex is either excitatory (probability $p$) or inhibitory (probability $1-p$). We take the mean-field limit of $Z^N$, leading to a multivariate point process $\bar Z$. If $p\neq\tfrac12$, we rescale the interaction intensity by $N$ and find that the limit intensity process solves a deterministic convolution equation and all components of $\bar Z$ are independent. In the critical case, $p=\tfrac12$, we rescale by $N^{1/2}$ and obtain a limit intensity, which solves a stochastic convolution equation and all components of $\bar Z$ are conditionally independent.