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

TL;DR

This paper analyzes the stability and convergence of solutions to a nonlinear renewal equation, providing conditions for stability, rates of convergence, and implications for Hawkes processes, including a Central Limit Theorem.

Contribution

It offers new stability and convergence results for nonlinear renewal equations and applies these findings to establish a Central Limit Theorem for Hawkes processes.

Findings

Stability and convergence results around equilibrium solutions.

Quantitative rates of convergence to equilibrium.

Instability in critical and supercritical cases.

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.