Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

TL;DR

This paper analyzes the long-time behavior of fluctuations in the Kuramoto synchronization model, revealing linear asymptotics and non-self-averaging properties through spectral analysis of an associated SPDE.

Contribution

It provides a spectral analysis of the fluctuation SPDE in the Kuramoto model, establishing linear asymptotics and non-self-averaging of fluctuations.

Findings

Fluctuations exhibit linear behavior for large time.

The model's fluctuations are not self-averaging.

Spectral analysis reveals a Jordan decomposition of the evolution operator.

Abstract

We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered Kuramoto model are not self-averaging.