Stochastic resetting and hierarchical synchronization

Abstract

Stochastic resetting is

simple enough to be approached analytically, yet modifies stochastic processes

in a non-trivial way. This modification entails making it possible that the

stochastic process has to restart from its initial position at any point in time. In part I of the thesis we study the effect it has on the statistical properties of additive functionals of the Ornstein-Uhlenbeck process and Brownian motion. We are particularly interested in the change of the probabilities of rare events. The Kuramoto model has been used to model synchronization for decades, yet the effect of the underlying structure of the interactions has only recently received attention. In Part II of the thesis we study the effect of community structure in the interaction network analytically in two simple cases, namely, a hierarchical network and a two-community network. The community structure significantly enriches the model in both cases.