Mean-field limits for non-linear Hawkes processes with inhibition on a Erdős-Rényi-graph

Abstract

We study a multivariate, non-linear Hawkes process Z N on a q-Erdős-Rényi-graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1 − p). If p = 1 2 , we take the mean-field limit of Z N , leading to a multivariate point process Z. We rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z are independent. The fluctuations around the mean field limit converge to the solution of a stochastic convolution equation. In the critical case, p = 1 2 , we rescale by N 1/2 and discuss difficulties, both heuristically and numerically.