Extinction versus unbounded growth; Habilitation Thesis of the University Erlangen-N\"urnberg

TL;DR

This thesis explores the duality between certain Markov processes and stochastic processes exhibiting extinction or unbounded growth, demonstrating how this relationship aids in understanding invariant measures and convergence in interacting particle systems.

Contribution

It introduces three new examples applying the extinction versus unbounded growth duality principle to analyze invariant measures in interacting particle systems.

Findings

Duality principle helps determine invariant measures.

Convergence to equilibrium can be studied via dual processes.

New examples illustrate the application in particle systems.

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.