Stochastic systems with locally defined dynamics

Abstract: We study three different classes of models of stochastic systems with locally defined dynamics. Our main points of interest are the limiting properties and convergence in these models. The first class is the locally interactive sequential adsorption, or LISA, models. We provide the general LISA framework, show that several classes of well-understood models fall within the framework, such as Polya urn schemes and fragmentation processes. We study several particular new examples of LISA processes having the feature of scalability. We provide the sufficient conditions for the existence of limiting empirical measures, and prove a bound for the speed of convergence. The second class is Bit Flipping models, where we study a behaviour of a sequence of independent bits, each flipping between several states at a given rate p_k. We define two particular models, Binary Flipping and Damaged Bits, and find the conditions on the rates {p_k} at which the models switch from the transient to the recurrent behaviour; as well as provide bounds for moments of the recurrence time under a certain set of conditions in the recurrent case, and prove the central limit theorem. The third class is Random Exchange Models where a countable collection of agents are trading independent random proportion of their masses with neighbours in a stepwise fashion. We find the stationary regimes for such models, and prove a limit theorem. As a corollary, we obtain a new invariance property of a stationary Poisson process on the real line with respect to a certain neighbour-dependent point shift.