Journal
ANNALS OF APPLIED PROBABILITY
Volume 32, Issue 1, Pages 360-391Publisher
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/21-AAP1680
Keywords
Interacting oscillators; long time dynamics; random graphs; stochastic partial differential equations; cut-norm; Grothendieck's inequality; self-normalized processes
Categories
Funding
- European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant [665850]
Ask authors/readers for more resources
This article analyzes a stochastic Kuramoto model defined on a sequence of graphs and investigates the relationship between the mean field limit, the connectivity of the underlying graph, and the long-time behavior. The research shows that under certain conditions, the empirical measure of the system can approach the solution of the equation associated with the classical mean field limit within finite time, even considering the network dependency. Furthermore, the study explores the long-time behavior and suggests that the empirical measure gradually approaches stable stationary solutions.
The stochastic Kuramoto model defined on a sequence of graphs is analyzed: the emphasis is posed on the relationship between the mean field limit, the connectivity of the underlying graph and the long time behavior. We give an explicit deterministic condition on the sequence of graphs such that, for any finite time and any initial condition, even dependent on the network, the empirical measure of the system stays close to the solution of the McKean-Vlasov equation associated to the classical mean field limit. Under this condition, we study the long time behavior in the subcritical and in the supercritical regime: in both regimes, the empirical measure stays close to the (possibly degenerate) manifold of stable stationary solutions, up to times which can diverge as fast as the exponential of the size of the system, before large deviation phenomena take over. The condition on the sequence of graphs is derived by means of Grothendieck's inequality and expressed through a concentration in l(infinity) -> l(1) norm. It is shown to be satisfied by a large class of graphs, random and deterministic, provided that the average number of neighbors per site diverges, as the size of the system tends to infinity.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available