### 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 **R**^{d} 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.