### Selected topics in Probability Theory:

Random walks on graphs

Monday 09.45-11.15, Seminarraum 9 (OMP 1)

First lecture on 09.03.2015

**Summary:** The lecture focuses on
study of random walks on (weighted) graphs and, in more
general, on study of general reversible Markov chains.

The following topics will be covered:

- random walks and eletrical networks
- connections to potential analysis
- harmonic functions on graphs
- estimates on transition density (with links to homogenisation theory)
- scaling limits of the random walk among random conductances, a particular type of random walk in random environment.

**Literature:**

- D. Aldous, J. Fill: Reversible Markov chains and random walks on graphs
- M. Biskup: Recent progress on the Random Conductance Model, Probability Surveys Vol. 8 (2011)
- T. Kumagai: Random Walks on Disordered Media and their Scaling Limits, Lecture Notes in Mathematics 2101, Springer (2014)
- R. Lyons, Y. Peres: Probability on Trees and Networks
- L. Saloff-Coste: Lectures on Finite Markov Chains, LNM 1665.

**Lecture notes:**

- March 9
- March 16
- March 23
- April 13
- April 20
- April 27
- May 4
- May 11
- May 18
- June 1
- June 8 - To be written
- June 15 - see Sections 2.1, 2.2 of Biskup (2011)
- June 22 - see Chapter 3 of Biskup (2011)

### Seminar in Probability Theory

Monday 13.15-14.45, Seminarraum 9 (OMP 1)

First session: 9 Mar 2015

The topics of the seminar will be 'Fractal
Geometry'.

Detailed list of topics will be given on March 9

**Literature:**

- K. Falconer: The Geometry of Fractal Sets, Cambridge UP, 1985
- K. Falconer: Fractal Geometry - Mathematical Foundations and Applications , John Wiley, Third Edition, 2014

**Programm:**

- 23.3. Box counting dimension (Krasser)
- 13. and 20.4. Haussdorff measure and dimension (Rydholm/Fischer)
- 27.4. Products of Fractals (Grocholski)
- 4.5. Iterated function systems (Bazant)
- 11.5. Random fractals, fractal percolation (Oucherif)
- 18.5. Graphs of functions (Breitner)
- 1.6. Ch. 10 or 14 (Sedlak)
- 7.6. Brownian motion (Cutraro)
- 14.6. ?? (Grass)