Abstract

The dynamics of a complex physical, biological, or chemical systems can often be modelled in terms of a continuous-time Markov process. The governing equations of these processes are the Fokker-Planck and the master equation. Both equations assume that the future of a system depends only on its current state, memories of its past having been wiped out by randomizing forces. Whereas the Fokker-Planck equation describes a system that evolves continuously from one state to another, the master equation models a system that performs jumps between its states. In this thesis, we focus on master equations. We first present a comprehensive mathematical framework for the analytical and numerical analysis of master equations in chapter I. Special attention is given to their representation by path integrals. In the subsequent chapters, master equations are applied to the study of physical and biological systems. In chapter II, we study the stochastic and deterministic evolution of zero-sum games and thereby explain a condensation phenomenon expected in driven-dissipative bosonic quantum systems. Afterwards, in chapter III, we develop a coarse-grained model of microbial range expansions and use it to predict which of three strains of Escherichia coli survive such an expansion.