Random walks in dynamic random environments are random walks evolving according to a random transition kernel, i.e., their transition probabilities depend on a stochastic process called dynamic... Show moreRandom walks in dynamic random environments are random walks evolving according to a random transition kernel, i.e., their transition probabilities depend on a stochastic process called dynamic random environment. In this thesis, we study asymptotic properties of such random walks on the integer lattice, in which the dynamic random environment is given by an interacting particle system. Show less
This thesis deals with Markov operators and semigroups. A Markov operator is a positive linear operator on the space of finite measures on some state space that preserves mass. A Markov semigroup... Show moreThis thesis deals with Markov operators and semigroups. A Markov operator is a positive linear operator on the space of finite measures on some state space that preserves mass. A Markov semigroup is a family of Markov operators parametrised by the positive real numbers, satisfying the semigroup property. These appear naturally in various places: deterministic dynamical systems, iterated function systems, structured population models and more generally Markov chains and Markov processes. We will study general Markov operators and semigroups in a functional analytic framework. Because the usual topology on the space of measures, given by the total variation norm, is often too strong for applications, we consider weaker topologies on the space of measures. and study continuity properties of Markov semigroups and their restriction to invariant subspaces. In the latter part of the thesis we provide ergodic decompositions, yielding, among other things, a characterisation of ergodic measures and an 'explicit' integral decomposition of invariant measures into ergodic measures. Under extra equicontinuity assumptions the ergodic decompositions have some nice properties, allowing us to find various characterisations for the existence, uniqueness, mean ergodicity and stability of invariant measures, and giving us extra information on the set of ergodic measures. Show less
The concept of distance is a fundamental notion that forms a basis for the orientation in space. It is related to the scientific measurement process: quantitative measurements result in numerical... Show moreThe concept of distance is a fundamental notion that forms a basis for the orientation in space. It is related to the scientific measurement process: quantitative measurements result in numerical values, and these can be immediately translated into distances. Vice versa, a set of mutual distances defines an abstract Euclidean space. Each system is thereby represented as a point, whose Euclidean distances approximate the original distances as close as possible. If the original distance measures interesting properties, these can be found back as interesting patterns in this space. This idea is applied to complex systems: The act of breathing, the structure and activity of the brain, and dynamical systems and time series in general. In all these situations, optimal transportation distances are used; these measure how much work is needed to transform one probability distribution into another. The reconstructed Euclidean space then permits to apply multivariate statistical methods. In particular, canonical discriminant analysis makes it possible to distinguish between distinct classes of systems, e.g., between healthy and diseased lungs. This offers new diagnostic perspectives in the assessment of lung and brain diseases, and also offers a new approach to numerical bifurcation analysis and to quantify synchronization in dynamical systems. Show less
