Large Deviations

This bachelor seminar took place in Bonn in the winter semester 2019/2020.

S2F2 - Hauptseminar Stochastische Prozesse und Stochastische Analysis: Großen Abweichungen – WS 2019/2020

Massimiliano Gubinelli / Nikolay Barashkov / Immanuel Zachhuber

Zeit und Ort: Mittwoch 10-12 SR 0.008

Die Theorie der groï£¡en Abweichungen behandelt in systematischen Weise die Berechnung von Wahrscheinlichkeiten “exponentiell unwahrscheinlicher” Ereignisse. Diese Theorie ist zu einem der wichtigsten Instrumente der Wahrscheinlichkeitstheorie geworden und erlaubt die Behandlung zahlreicher Anwendungsprobleme. In dem Seminar wollen wir die wichtigsten Grundlagen dieser Theorie erarbeiten und auch einige interessante Anwendungen kennenlernen. Grundlage bildet das Buch “A Weak Convergence Approach to the Theory of Large Deviations", von Dupuis, Paul, and Richard S. Ellis.

Vorkenntnisse. Mindestens Einfï£¡hrung in die W-Theorie, ein bisschen auch noch Stochastische Prozesse

The theory of large deviations systematically deals with the calculation of probabilities of “exponentially unlikely” events. This theory has become one of the most important tools of probability theory and allows the treatment of many application problems. In the seminar we want to elaborate the most important basics of this theory and get to know some interesting applications.

Previous knowledge: At least introduction to probability theory, better also stochastic processes

Reference text: Dupuis, Paul, and Richard S. Ellis. 1997. A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York. https://doi.org/10.1002/9781118165904.
Presentation of the seminar (slides)

Tentative planning

(9/10) Large deviations in terms of Laplace principle (1.1-1.2) [Yuanxi Zhang]
(23/10) Basic results in the theory (1.3) [Nikolay Barashkov]
(30/10) Properties of relative entropy (1.4) [Massimiliano Gubinelli]
(6/11) Γ-convergence and Gibbsian-conditioning (notes) [Massimiliano Gubinelli]
(13/11) Sanov's theorem. Statement and representation formula (2.1-2.3) [Hannah Westhoff]
(20/11) Lower and upper bounds (2.4-2.5) [Mattia Turra]
(27/11) Mogulskii's theorem. Representation formula (3.1-3.2) [Immanuel Zachhuber]
(11/12) Upper bound and rate function (3.3) [Luigi Borasi]
(18/12) Statement of the theorem and proof + Cramï£¡rs theorem + comments (3.4-3.5-3.6)
(8/1) Random walk model, rep formula + compactness (5.2-5.3)
(15/1) Upper bound and rate function (6.2)
(22/1) Lower bound and statement of the theorem (6.5)
(?) Markov chains, rep formula + compactness (8.2)
(?) Upper bound and rate function (8.3-8.4)
(?) Properties of rate function and Lower bound (8.5-8.6)