Random walks in random environment
Alejandro Ramirez
Winter term, 2014-2105
Institut für Mathematik - Technische Universität Berlin 


This course will cover some aspects of non-reversible random walks in random enviroments moving on the hypercubic lattice Z^d. Students should have a knowledge of probability theory and stochastic calculus. It will be based on the following lecture notes

"Selected topics in random walks in random environment" A. Drewitz and A. Ramirez, Topics in percolative and disordered systems, A. Ramirez, G. Ben Arous, P. Ferrari, C. Newman, V. Sidoravicius, M.E. Vares (editors), Springer Proceedings in Mathematics and Statistics, 69, 23-83 (2014).

The schedule is on tuesdays and thursdays between 10:00 and 11:30. Nevertheless on tuesday october 14, it might be possible, but difficult to modify it if students wish to do it.

I. The environmental process and its invariant measures.
1. Definitions.
2. Invariant probability measure of the environment as seen from the random walk.
3. Transience and recurrence in the one-dimensional model.
4. Computation of an absolutely continuous invariant measure in dimension d=1.
5. Absolute continuous invariant measures and some implications.
6. The law of large numbers, directional transcience and ballisticity.
7. Transience, recurrence and a quenched invariance principle.
8. One-dimensional quenched large deviations.
9. Multidimensional quenched large deviations.
10. Variational formula for the multidimensional quenched rate function.

II. Trapping, ballistic behavior and other topics (higher chances of modifying the program in this chapter).
1. Directional transcience.
2. Renewal structure.
3. Law of large numbers.
4. Ballisticity.
5. Ellipticity conditions for ballistic behavior.
6. Random walk in Dirichlet random environment.
7. Connections with reinforced random walks.

The seminar "Topics in random media" will cover some articles related to the content of this course. More information can be found here.

Additional bibiliography