# MA946 - Introduction to Graduate Probability

**Lecturer: **Stefan Adams

** Term 1:** Three lectures per week are scheduled for f2f teaching in MS.01 (large lecture theatre with a capacity of 300 allowing for social distancing): Monday 16--18 h and Wednesday 12--13 h. If the situation changes lectures will be online.

One additional online hour for exercises and discussions: Thursday 11--12 h.

**Commitment:** 30 lectures

**Assessment:** Oral exam (50%) 08.01.2021 (guidelines), Essay (50%) (guidelines)

**Essay hand-in time: (****hardcopy & pdf-file) by Friday 15 January 2021 at 12 pm.**

**Lecture Notes: to be updated on regular basis (pdf); Appendices (pdf)**

**Thursday online sessions (files): 8 October 2020 (pdf); 15 October 2020 (pdf); 22 October 2020 (pdf) & Literature (pdf); 29 October 2020 (pdf); 5 November 2020 (pdf); 12 November 2020 (pdf); 19 November 2020 (pdf); 27 November 2020 (pdf); 3 December 2020 (pdf); 4 December 2020 (pdf); 7 December 2020 (pdf); 9 December 2020 (pdf); 10 December 2020 (pdf) **

**Prerequisites: **Familiarity with topics covered in ST111 Probability A \& B; MA258 Mathematical Analysis III or MA259 Multivariate Calculus or ST208 Mathematical Methods or MA244 Analysis III; some MA359 Measure Theory or ST342 Maths of Random Events is useful.

The purpose of this module is to provide rigorous training in probability theory for students who plan to specialise in this area or expect probability to feature as an essential tool in their subsequent research. It will also be accessible to students who never got into probability theory beyond the core-module level taught in the first year and who are eager to get acquainted with basic probability theory, in particular, the aim is to appeal to but not limited to students working in analysis, dynamical systems, combinatorics & discrete mathematics, and statistical mechanics. To include these two different groups of students and to accommodate their needs and various background the module will cover in the first two weeks a steep learning curve into basic probability theory (see part I below). Secondly, the written assessment, 50 % essay with 16 pages, can be chosen either from a list of basic probability theory (standard textbooks in probability and graduate lecture notes on probability theory) or from a list of high-level hot research topics including original research papers and reviews and lecture notes (see below). List of possible essay topics (pdf).

**Content:**

**Part I: Introduction to basic probability theory (week 1-3)**

- Random variables, distributions, and convergence criteria

- Law of large numbers

- The Central Limit Theorem

- Markov processes (random walks in discrete-time)

**Part II: Introduction to core areas in probability theory (week 4-8)**

The aim will be to develop problem-solving skills together with a deep understanding of the main ideas and techniques in probability theory in the following core areas during the following 5-6 weeks:

- Large deviation theory (Cramer and Sanov theorem; Varadhan Lemma; Schilder's theorem; basic principles, and applications

- Brownian Motion (definition and construction; Blumenthalâ€™s 0-1 Law; Donsker's theorem; local times; Wiener measure; Classical Potential theory).

**Part III: Optional topics and overview (week 9-10)**

The third aim and part of the lecture in the remaining weeks will be to provide an overview of important areas of modern probability.

- Gaussian Free Field (definitions, Gibbs measures, random walk representation, continuum limits)

If time permits in week 10 the lecture provide an introduction to Wasserstein gradient flow and large deviation theory

**References:
**

Hans-Otto Georgii, Stochastics, De Gruyter Textbook, 2nd rev. and ext. (2012).

Peter Moerters and Yuval Peres: Brownian motion, Cambridge University Press (2010).

Daniel W. Stroock: Probability - An analytic view; revised ed. Cambridge University Press (1993).

Olav Kallenberg: Foundations of Modern Probability, 2nd ed. Springer (2002).

L.C.C. Rogers & D. Williams: Diffusions, Markov processes, and martingales Vol 2, Cambridge University Press (2000).

Daniel W. Stroock & S.R. Srinivasa Varadhan: Multidimensional Diffusion Processes, Springer (1979).

Amir Dembo and Ofer Zeitouni: Large Deviations Techniques and Applications, Springer (1997).

Frank den Hollander, Large Deviations (Fields Institute Monographs), (paperback), American Mathematical Society (2008).

Jin Feng and Thomas G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society (2006).

Gregory Lawler & Vlada Limic: Random Walk: A Modern Introduction, Cambridge University Press (2000).