Random walks in random environment
Alejandro Ramirez
Winter term, 2014-2105
Institut für Mathematik
- Technische Universität Berlin
NOTICE: STARTING ON TUESDAY OCTOBER 21, CLASSES WILL BE IN ROOM MA748.
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
- F. Rezakhanlou. "A prelude to the theory of random walks in
random environments". Bull. Iranian Math. Soc. 37(2), 5-20
(2011).
- A. S. Sznitman. "Topics in random walks in random
environment". School and conference on probability theory,
ICTP Lect. Notes, XVII, 203-266, (2004).
- O. Zeitouni. "Random walks in
random environment". XXXI Summer school in probablility,
St. Flour (2001). Lecture Notes in Math., vol. 1837, pp.
193-312. Springer, Berlin (2004).