Universität Duisburg-Essen

Probability Group at Campus Essen

Wolfgang Löhr and Anita Winter

26th March 2013

Course (4SWS) with integrated problem sessions/Seminar presentations (2SWS)

Summer Semester 2013

Tuesday and Thursday, 10.15-12.00 (Room: WSC-S-U-3.03)

Summer Semester 2013

Tuesday and Thursday, 10.15-12.00 (Room: WSC-S-U-3.03)

**Markov processes: Theory and Examples**

*Content:*
The development of any stochastic model involves the identification of properties and parameters that, one hopes, uniquely characterize the stochastic process.
In fact, the model may be arrived by some sort of limiting or approximation argument.
For example, Brownian motion arises as the suitably re-scaled limit of random walks (with small but frequent jumps).
To be in a position to state such a convergence we develop concepts such as the weak topology in the space of probability measures and the
Skorokhod topology in the space of cadlag paths.
In this class we want to focus on the theory of Markov processes which are an important class of stochastic processes.
We also introduce operator semigroups, martingale problems, stochastic equations which provide approaches to the characterization of Markov processes.
The theory will be illustrated with many examples.
The course is accompanied by a block seminar in which students can present a paper related to the theory.

- Topological and probabilistic preliminaries
- Cadlag paths and the Skorokhod metric
- Markov processes
- Weak convergence and the Prohorov metric
- Feller semigroup and generators
- Harmonic functions and martingales
- Convergence of stochastic processes
- Strong Markov property
- Martingale problems and Dirichlet forms

*Präsentationen*

- 9.00-9.45 Strong Markov property, Mussarrat Mouri, Script: Section 7
- 9.45-10.30 Martingale Problem Formulation I, Oliver, Script: Section 5
- 10.50-11.35 Martingale Problem Formulation II, Valentin, Script: Section 5
- 11.35-12.20 A tightness criterion, Roland Maizis, Script: Section 6
- 12.30-13.15 Duality; Examples, Maurice Babnik
- 14.30-15.15 Convergence of Galton-Watson processes to Feller diffusion, Qian
- 15.15-16.00 Stationary measures, Gerrit
- 16.20-17.05 Construction of the contact process, Tim Schönberger, Liggett:
- 17.15-18.00 Construction of the exclusion process, Johannes Lankeit, Seppäläinen: Section 2

*Preknowledge:*
The course is offered in the Master program in Mathematics. It is addressed to students with strong knowledge in measure-theoretic probability theory (Wahrscheinlichkeitstheorie I), preferable
Wahrscheinlichkeitstheorie II and/or Funktionalanalysis. Further knowledge about stochastic processes are of advantage. We will adapt the class to the background of the participants.
To attract non-German speaking students we are planning to teach this course in English.
If all students are comfortable with German, we will teach in German.
In any case we mainly follow a script which is written in English.
If your are interested in the course, please send an e-mail
(anita.winter@uni-due.de)
and indicate your pre-knowledge and whether or not it is ok for you to follow the class in English.

*Literature:*

- Steward Ethier and Thomas Kurtz: Markov processes: Characterization and Convergence, 1986.
- Thomas Liggett: Continuous Time Markov Processes (Graduate Studies in Mathematics), 2010
- Timo Seppäläinen: Translation Invariant Exclusion Processes, 2003, (Book in Pogress)
- Jan Swart and Anita Winter: Lecture notes. Script