Information to seminar: Random Walks (WS 2025/2026)
Lecturer: Prof. Dr. Angelika Rohde
Lecture assistant: Dr. Johannes Brutsche
Date/Location: Mo, 16-18 Uhr, SR 127, Ernst-Zermelo-Str. 1
Kick-off meeting: 22.07., 14:00, room 232, Ernst-Zermelo-Str. 1
Language: preferable English
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Content
Random walks are stochastic processes (in discrete time) formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. Many results that are part of this seminar also carry over to Brownian motion and related processes in continuous time. In particular, the theory for random walks contains many central and elegant proof ideas which can be extended to various other settings. We start the theory at the very beginning but quickly move on to proving local central limit theorems, study Green's function and recurrence properties, hitting times and the Gambler's ruin estimate. Further topics may include a dyadic coupling with Brownian motion, Dirichlet problems, random walks that are not indexed in ℕ but in the lattice ℤd, and intersection probabilities for multidimensional random walks (which are stochastic processes from ℕ to ℝn). Here, we will see that in dimension n=1,2,3 two paths hit each other with positive probability, while for n>3 they avoid each other almost surely.
Schedule
| Date | Title |
| 20.10.2025 | Introduction |
| 27.10.2025 | Green's function (part 1) |
| 03.11.2025 | Green's function (part 2) |
| 10.11.2025 | One-dimensional random walks |
| 17.11.2025 | Potential theory |
| 24.11.2025 | Loop measures |
| 01.12.2025 | Local CLTs (part 1) |
| 08.12.2025 | Local CLTs (part 2) |
| 12.01.2026 | Intersection probabilities |
There may follow further talks that are not scheduled yet. In case you are interested in the seminar, please reach out to the assistant.
Literature
- Gregory F. Lawler, Vlada Limic, Random walk - a modern introduction. Cambridge studies in advanced mathematics 123. (https://www.math.uchicago.edu/~lawler/srwbook.pdf)
Prerequisites
Probability Theory I.
Some talks only require knowledge in Stochastics I. In case you have not yet attended the lecture in Probability Theory, please reach out to the assistant and discuss which topics might be suitable for you.
Usability
Wahlmodul im Optionsbereich (2HfB21)
Mathematisches Seminar (BSc21)
Wahlpflichtmodul Mathematik (BSc21)
Mathematische Ergänzung (MEd18)
Mathematisches Seminar (MSc14)
Wahlmodul (MSc14)
Mathematical Seminar (MScData24)
Elective in Data (MScData24)
4a. EUCOR-Partneruniversitäten
Im Rahmen der EUCOR-Kooperation können Sie Veranstaltungen an den Partnerhochschulen besuchen. Wenn Sie auf die einzelnen Universitäten klicken, finden Sie Links zu deren Vorlesungsverzeichnissen.
Sprechstunden
Office hour lecturer: Tuesday 4-5 pm.
Office hour assistant: Tuesday 2-3 pm, or by appointment.