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

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 Z & macr;. If p&NOTEQUexpressionL;1/2, we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z & macr; are independent. In the critical case, p = 1/2, we rescale by N-1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z & macr; are conditionally independent. (C) 2022 Elsevier B.V. All rights reserved.