{
  "id": "2102.01052",
  "version": "v1",
  "published": "2021-02-01T18:42:14.000Z",
  "updated": "2021-02-01T18:42:14.000Z",
  "title": "Mean-field limits for non-linear Hawkes processes with excitation and inhibition",
  "authors": [
    "Peter Pfaffelhuber",
    "Stefan Rotter",
    "Jakob Stiefel"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T18:42:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60F05"
    ],
    "keywords": [
      "non-linear hawkes process",
      "mean-field limit",
      "inhibition",
      "excitation",
      "limit intensity process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}