Extinction versus unbounded growth; Habilitation Thesis of the University Erlangen-Nürnberg

Abstract

Certain Markov processes, or deterministic evolution equations, have the property that they are dual to a stochastic process that exhibits extinction versus unbounded growth, i.e., the total mass in such a process either becomes zero, or grows without bounds as time tends to infinity. If this is the case, then this phenomenon can often be used to determine the invariant measures, or fixed points, of the process originally under consideration, and to study convergence to equilibrium. This principle, which has been known since early work on multitype branching processes, is here demonstrated on three new examples with applications in the theory of interacting particle systems.