Topics in random walks, Summer Semester 2019

Lecture. Monday, 14:00-16:00, room MA 648 (TU Berlin math building).

Dates. April 8 to July 8. No lectures on April 22 and June 10 (public holidays).

Language. English.

Prerequisites. Stochastics I is mandatory, Stochastics II is preferable.

Course number. 3236 L 321.

Exams (oral). July 22, 2019 10:00-16:00 and July 23, 2019 9:00-15:00.

Contact. Tal Orenshtein, orenshtein(at)tu-berlin.de, room MA 784, TU Berlin.

Office hours. By appointment.

Lecture notes.

Recommended literature.

• Steven Lalley's Lecture notes on one-dimensional simple random walks.
• Ron Peled's course "Random walks and Brownian motion".
• Martin Hairer's lecture notes on convergence of Markov processes.
• The book "Non-homogeneous random walks" by Menshikov, Popov and Wade. Old version is available online in Serguei Popov's webpage.
• Jonathon Peterson's lecture notes on random walks in random environment.
• Ofer Zeitouni's lecture notes on random walks in random environment.
• Review paper by Elena Kosygina and Martin Zerner on excited random walks.

Content.

Lecture 1. Basic properties of SRW on Z: computing the expected time to leave a finite interval and the probabilities to leave it in a certain side, recurrence, stopping times.

Lecture 2. Strong Markov property, skip-free walks, cycle lemma and first passage time density asymptotics, the running maximum and the reflection principle.

Lecture 3. Ballot theorem, Wald identities for RW on Z with general jumps, sample path properties of SRW on Z - main lemma.

Lecture 4. SRW on Z: Arc-sine laws for the last zero and for positive time, time reversal property and more arc-sine laws. equidistribution of SRW bridge positive time, Local CLT, Hoeffding's inequality and formulation of law of iterated logarithm.

Lecture 5. Recurrence, transience and the Green function, Polya's theorem for SRW on Z^d. Exchangeable events and Hewitt-Savage 0-1 law.

Lecture 6. RW on R^d: point and neighborhood recurrence and the Green function, formulation of Chung-Fuchs theorem. Corollaries of Birkhoff's ergodic theorem: LLN and Kesten-Spitzer-Whitman's theorem on the range of RW. Application to the escape probability of biased SRW on the integers.

Lecture 7. Harmonic functions, the maximum principle and the Dirichlet problem for RW on graphs. Random walks in random environment (RWRE) - definition and some examples.

Lecture 8. Solomon's recurrence/transience criterion for RWRE on the integers.

Lecture 9. Solomon's SLLN and ballicticity criterion for RWRE on the integers.

Lecture 10. Asymptotic behavior of countable Markov chains: Foster-Lyapunov criterion for transience/recurrence/positive-recurrence. Immediate applications: SRW on the naturals, with or without a drift.

Lecture 11. Survival/extinction criterion for branching processes with (random) migration (BPwM).

Lecture 12. Excited random walks (ERW) and their relation to BPwM. An application: recurrence/transience criterion for ERW.