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... Show moreStochastic 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. Show less
This PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the... Show moreThis PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the classical Hilbert space setting to the setting of general CAT(0) spaces. In the third chapter, the non-symmetric Fokker-Planck equation is treated as a flow on the Wasserstein-2 space of probability measures, and it is proven that its semigroup of solutions possesses similar properties to those of the gradient flow semigroups. In the forth chapter, a general theory of maximal monotone operators and the induced flows on Wasserstein-2 spaces is developed. This theory generalizes the theory of gradient flows by Ambrosio-Gigli-Savare. In the final fifth chapter the problem of the existence of an invariant measure for stochastic delay equations is proven. The diffusion coefficient has delay, and is assumed to be locally Lipschitz and bounded. Show less
