Math 778 — Spring 2001 Stochastic Processes: Random Walks on Finitely Generated Groups
The aim of this course is to present some of the basic results concerning random walks on finitely generated groups. This means the emphasize will be on non-abelian groups and not on the much more advanced and refined theory of random walks on Euclidean lattices and abelian groups presented in Spitzer's book.
For instance, we will prove that for polycyclic groups, the probability of return of a simple random walk to its starting point after n steps decay either as the reciprocal of a polynomial (if the group is virtually nilpotent) or as the exponential of minus the third root of n (if the group has exponential volume growth). We will spend some time discussing simple random walks on nilpotent groups in detail.
No specific background is required. Many different areas of mathematics will come into play (algebra, analysis, geometry, probability theory) and some difficult results will be explained and used without proof, but many of the main arguments will be of an elementary nature. To some extend, interest and the willingness to use many different areas of mathematics is all is required from a graduate student to follow the course.
Here is a tentative list of topics to be discussed:
Simple random walks, Cayley graphs, volume growth, amenability, transience, Nash inequalities, harmonic functions, rate of escape, number of visited points, ...
There will be a set of notes covering parts of the course. There is a also a recent book: W. Woess, Random Walks on Infinite Graphs and Groups.