Lecture

Random Dynamical Systems

This course will be examined via an oral examination of 20 minutes.
The oral examination concerns the material presented in the lectures, which itself is based on detailed lecture notes that will be made available. The students will receive guidance on how to
prepare for the oral examination towards the end of the course.

Abstract: 
The development of dynamical systems theory is one of the scientific revolutions of the 20th century, and many important insights from this field are now embedded within computational methodologies in many branches of science and engineering as well as at the heart of deep and abstract mathematics. During the last few decades, the importance of noise and uncertainty has become evident in real-world applications of dynamical systems, but a corresponding random dynamical systems theory is still only in its very early stages of development. The John-von-Neumann lectures will focus in the first instance on fundamental aspects of the theory, as initiated in the 1980s and 1990s by the Bremen group around Ludwig Arnold, but recent developments will be covered as well.

1) Ergodic theory. Due to its importance for the random dynamical systems theory, the first part of the course focusses on ergodic theory, which concerns the statistical description of dynamical
systems. We cover invariant measures, Poincaré recurrence, Krylov-Bogolubov theorem, ergodicity, Birkhoff ergodic theorem, as well prototypical examples.

2) Basic properties of random dynamical systems. We cover fundamental definitions and explain how random dynamical systems are generated in discrete time (via random difference equations)
and in continuous time (via random and stochastic differential equations). We also explain how Markov random dynamical systems relate to Markov processes.

3) Invariant measures for random dynamical systems. There are two types of invariant measures of interest: stationary measures (that give statistical information) and random invariant measures
(that contain more dynamical information). We discuss existence of these measures in specific contexts and how they relate to each other (correspondence theorem).

4) Attractors for random dynamical systems. Different notions of attraction have been studied for random dynamical systems, and we cover pullback, forward and weak attraction. We provide insights
into existence and uniqueness of attractors and guidance on how to prove the existence of attractors in specific situations.

5) Synchronisation in random dynamical systems. We explore the phenomenon of synchronisation, which describes convergence of different trajectories when driven by the same noise realisation.
We focus on specific situations (e.g. random circle mappings) and also discuss the phenomenon of destruction of bifurcations in the stochastic differential equations context when using unbounded noise.

6) Multiplicative ergodic theory. The existence of a Lyapunov spectrum for linearisations of random dynamical systems along trajectories is provided by the multiplicative ergodic theorem. We
provide a proof of this celebrated result in dimension two, and we cover the existence of a top Lyapunov exponent in arbitrary dimension via the Furstenberg-Kesten theorem and Kingman's subadditive ergodic theorem.

7) More decompositions of random dynamical systems. The global dynamical behaviour of random dynamical systems on compact phase spaces is understood by its Morse decompositions. We provide
insights into the existence of such decompositions by using weak attractors-repeller pairs and discuss its dynamical implications.

8) Bounded noise random dynamical systems. We address certain shortcomings of unbounded noise (such as destruction of bifurcations) and develop an understanding of the topological
properties of bounded noise random systems and their bifurcations.

The written course material will be in English; the lectures will be held in English.

Target audience:
- knowledge of fundamental as well as advanced aspects of the random dynamical systems theory.
- knowledge of how various techniques from different fields of mathematics are used in the context of random dynamical systems.

Literatur: 
Parts of the course material can be found in the following three books:
- L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, New York, 1998.
- I. Chueshov, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics 1779, Springer, 2002.
- S. Kuksin and A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics, vol. 194, Cambridge University Press, Cambridge, 2012.

A complete list of references with papers containing certain results that are covered in the lectures will be made available in the lecture notes.

Scheduled Dates

ECTS: 3 CP

Prerequisites (referring to courses at TUM): Knowledge on measure and integration theory is required; in various places, we also make use of techniques from dynamical systems,
ergodic theory, stochastic processes, and functional analysis. Knowledge of these fields is helpful but not required: the lecturer will provided sufficient background information during the lectures
to make the lectures as self-contained as possible.