Summary

This course will give an overview how stochastic processes are seen from a viewpoint of dynamical systems theory. For deterministic systems that, for example, exhibit chaotic properties such that predictions for single trajectories are impossible, it is often very useful to make statements about the probability distribution of many trajectories. This observation is the basis of ergodic theory which focuses on probability distributions that stay invariant under the dynamics and correspond to asymptotic time averages of typical trajectories.

We will study systems where the randomness is part of the dynamics, for example in the form of stochastic differential equations, and discuss the connections to ergodic theory. Random dynamical systems is the mathematical theory for many real-world phenomena, such as synchronization or chaos, encountered in statistical and quantum physics, climate science, molecular dynamics, finance and economics and many others. 

Organization

Lecture:  Tue 8:15am -10 am (15 minutes break in between), First Letcure: 13th of April.
Exercise class: Mon 12:15am - 13:45 pm, First Exercise class: 19th of April.

The lecture and the exercise classes will be held online via Webex. The corresponding links are posted in the Announcements. Updated lecture notes and exercise sheets will be shared via Resources.

Literature

L. Arnold, “Random Dynamical Systems”, Springer, 1998 (2^nd printing, 2003).

H. Crauel and P.E. Kloeden, “Nonautonomous and Random Attractors”, Jahresber Dtsch Math-Ver (2015) 117:173–206.

Y. Kifer, "Ergodic Theory of Random Transformations", Birkhäuser 1986.

S. Kuksin and A. Shirikyan, "Mathematics of two-dimensional turbulence", Cambridge University Press, 2012.

P.D. Liu and M. Quian, “Smooth Ergodic Theory of Random Dynamical Systems”, Springer, 1995.

Z. Schuss, “Theory and Applications of Stochastic Processes. An Analytical Approach”, Springer, 2010. (One possible reference for background on Stochastic Processes and SDEs)