Stochastic systems with locally defined dynamics

Licentiate thesis, 2012

This thesis considers two large classes of models related to the dynamical point processes. The first is the locally interactive sequential adsorption, or LISA, models. We provide the general LISA framework, show that a lot of well- understood models can be described within the framework, such as Polya urn schemes, fragmentation processes, cooperative sequential adsorption. We study several particular new examples of LISA processes which possess the feature of scalability. Our results describe the limiting behaviour of empirical measures of such processes. The second class is Bit Flipping models, where we study a behaviour of a sequence of independent bits, each flipping between several states at given intensity p_k. We investigate conditions on p_k at which the model switches from transient to recurrent behaviour, prove the central limit theorem for the transient case, and provide a bound for moments of the recurrence time in the recurrent case.