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Peer-reviewedCC BY 4.0Nonlinearity

Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes

  • Céline Duval
  • Eric Luçon
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Abstract

We study the asymptotic properties of the solutions of a nonlinear renewal equation. The main contribution of the present article is to provide stability and convergence results around equilibrium solutions, under some local subcritical condition. Quantitative rates of convergence to equilibrium are established. Instability results are given in both the critical and supercritical cases. As an implication of these results, we establish a Central Limit Theorem for Hawkes processes in a mean-field interaction.

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Cite

BibTeXRIS
BibTeX
@article{duval2026local,
  title = {{Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes}},
  author = {Céline Duval and Eric Luçon},
  journal = {Nonlinearity},
  year = {2026},
  month = sep,
  doi = {10.1088/1361-6544/aea8b8},
  eprint = {2512.14214},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05416121v2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes
AU  - Céline Duval
AU  - Eric Luçon
PY  - 2026
DA  - 2026/09/29
JO  - Nonlinearity
DO  - 10.1088/1361-6544/aea8b8
UR  - https://hal.science/hal-05416121v2
AB  - We study the asymptotic properties of the solutions of a nonlinear renewal equation. The main contribution of the present article is to provide stability and convergence results around equilibrium solutions, under some local subcritical condition. Quantitative rates of convergence to equilibrium are established. Instability results are given in both the critical and supercritical cases. As an implication of these results, we establish a Central Limit Theorem for Hawkes processes in a mean-field interaction.
SN  - 0951-7715
SN  - 1361-6544
ER  -