The goal of this student seminar is to understand some properties of random walks, especially on non-commutative groups.

Time: Tuesdays 3-4:20pm
Location: E 312

Some references

[FS] Alex Furman's Survey
[FRW] Furstenberg's Survey
[KL] Karlsson-Ledrappier
[KV] Kaimanovich-Vershik
[BQ] Benoist-Quint book

Provisional schedule

Date Speaker Topic
6 Jan 2015 Rachel Vishnepolsky Introduction. Recurrence, transience of RWs.
13 Jan 2015 Ian Frankel Law of Large numbers. Central Limit theorem.
20 Jan 2015 Bena Tshishiku Random Walks on groups. Markov operators.
27 Jan 2015 Clark Butler Poisson boundary for RWs. Harmonic functions.
3 Feb 2015 Mary He Furstenberg's theorem: Lebesgue measure is hitting
measure for a special RW on a lattice in a Lie group.
10 Feb 2015 Yun Yang Multiplicative Ergodic Theorem following Karlsson-Ledrappier I.
17 Feb 2015 Yun Yang Multiplicative Ergodic Theorem following Karlsson-Ledrappier II.
24 Feb 2015 Simion Filip Entropy. Zero entropy is equivalent to trivial boundary.
Shannon-McMillan-Breiman theorem. Differential Entropy.
3 Mar 2015 Paul Apisa Limit theorems in semisimple Lie groups:
LLN, CLT, Local CLT (following Benoist-Quint).
10 Mar 2015 Steve Lalley Positive entropy is equivalent to positive drift.
17 Mar 2015 Steve Lalley Examples: Lamplighter groups.
[2 lectures] Introduction.
Overview of the seminar. Some classical theorems.
[1 lecture] Strong and Weak Law of Large Numbers, Central Limit Theorem, Local Limit Theorem.
[1 lecture] Differences between \(\mathbb{Z}\) and \(\mathbb{Z}^d\). Recurrence, Green's function, Polya's theorem.
[3 lectures] Furstenberg's theorems, Poisson boundaries.
[1 lecture] Introduction to random walks on general groups. Markov Operators. Random Ergodic Theorem.
[1 lecture] Constructing the Poisson boundary. Poisson formula. (Martin Boundary?)
[1 lecture] Measures on lattices that give Lebesgue measure on the boundary. Applications.
[2 lectures] Noncommutative ergodic theorem.
Follow Karlsson-Ledrappier.
[1 lecture] Entropy.
Follow Kaimanovich-Vershik.
[1 lecture] Different notions of entropy. Speed of divergence.
                Another construction of Poisson boundaries. Boundary entropy.
[3 lectures] Applications.
[1 lecture] Limit theorems in semisimple Lie groups.
                Transfer operators, spectral gap, characteristic functions.
[1 lecture] Positive entropy is equivalent to positive drift.
[1 lecture] Some examples.