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- ItemDynamical phase transitions for flows on finite graphs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Gabrielli, Davide; Renger, D. R. MichielWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.
- ItemLarge deviations for Markov jump processes with uniformly diminishing rates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Agazzi, Andrea; Andreis, Luisa; Patterson, Robert I. A.; Renger, D. R. MichielWe prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics.