Brownian Motion and Stochastic Calculus (Spring 2011)

Summary: This is a first course on continuous-time stochastic processes. It covers basic notions of stochastic analysis. The following topics will be discussed:
- Brownian motion: definition, construction, some important properties
- Markov processes: basics, strong Markov property, generators and martingale problems
- Stochastic calculus: semimartingales, stochastic integrals, Ito formula, Girsanov transformation, stochastic differential equations
- Levy processes: basic notions, some important properties

Lecture notes: scanned handwritten notes

Large Deviations (Fall 2010)

Summary: The objective of the course is to gain familiarity with the methods of the large deviation theory and to learn some of its most important tools. We will cover at least theorems of Cramer (on Rd and on Polish spaces), Sanov, Gärtner-Ellis, and of Schilder, as well as Varadhan's lemma. If the time permits, some basics of the Freindlin-Wentzell theory will be introduced.

Lecture notes: scanned handwritten notes, collected exercise sheets

Percolation (Fall 2009)

Summary: The objective of the course is to gain familiarity with methods of the percolation theory and to learn some of its important results. Among others, we will show FKG and BK inequalities, Theorems of Harris-Kesten, Menshikov, Burton-Keane, and, if time permits, of Smirnov.

Knowledge of the probability theory on the level of the lecture 'Wahrscheinlichkeitstheorie' is more than sufficient for the course.

Lecture notes (without the last lecture): pdf

Markov Chains (Fall 2008)

Summary: The goal of the course is to introduce basic concepts of the theory of Markov chains, both in discrete and continuous time. Knowledge of the probability theory on the level of the lecture 'Wahrscheinlichkeitstheorie' is supposed.

Lecture notes: scanned handwritten notes (7MB)

Special Topics in Probability
- Aging in dynamics of disordered systems (Fall 2007)

Abstract: It was observed experimentally that certain real-world materials relax very slowly to equilibrium and, moreover, the relaxation manifests some unusual features like aging, rejuvenation and memory effects. The goal of the lecture is to present some recent mathematical results explaining some of these features in the context of Markovian dynamics in disordered systems, in particular in mean-field spin glasses. In order to achieve the goal, the lecture will include some more advanced chapters from the theory of extremes of i.i.d. random variables and of Gaussian processes, from the renewal theory and from the theory of Lévy processes. Some results on random walks on graphs will also be introduced.

The lecture should be essentially self-contained. Only basic knowledge of Markov chains and theory of i.i.d. random variables will be required.

Lecture notes: pdf

Wahrscheinlichkeitstheorie und Statistik für D-ITET (Spring 2009 and 2010)

Abstract: Wahrscheinlichkeitsmodelle und Anwendungen, Einführung in die Estimationstheorie und in die statistischen Tests.

Weitere Informationen: auf der Seite der Vorlesung und im Vorlesungsverzeichnis.