Prerequisite
Conditional expectation and Discrete Martingales
Introduction
Gaussian vectors.
Brownian motion: Construction, continuity, nondifferentiability.
Strong Markov property
Multidimensional BM: harmonic functions, recurrence and transience
Skorokhod embedding and the Law of the iterated logarithm
Stochastic integral and Ito formula with respect to BM
Conformal invariance of Brownian motion paths
Continuous time martingales and local martingales
Stochastic integration with respect to local martingales and semimartingales
General Ito formula
Girsanov theorem
Basic properties of Hausdorff dimension
Hausdorff dimension of the zero-set, path and graph of BM
Local time of BM
Intersections of BM paths
Exceptional sets on BM paths
Grading based on Homework, Class Participation, Midterm and Final Exam
Lecturer Intro
Yuval Peres obtained his PhD in 1990 from the Hebrew University, Jerusalem. He was a postdoctoral fellow at Stanford and Yale, and was then a Professor of Mathematics and Statistics in Jerusalem and in Berkeley. Later, he was a Principal researcher at Microsoft. In 2023, he joined Beijing Institute of Mathematical Sciences and Applications. He has published more than 350 papers in most areas of probability theory, including random walks, Brownian motion, percolation, and random graphs. He has co-authored books on Markov chains, probability on graphs, game theory and Brownian motion, which can be found at https://www.yuval-peres-books.com. His presentations are available at https://yuval-peres-presentations.com. He is a recipient of the Rollo Davidson prize and the Loeve prize. He has mentored 21 PhD students including Elchanan Mossel (MIT, AMS fellow), Jian Ding (PKU, ICCM gold medal and Rollo Davidson prize), Balint Virag and Gabor Pete (Rollo Davidson prize). He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, at the 2008 European congress of Math, and at the 2017 Math Congress of the Americas. In 2016, he was elected to the US National Academy of Science.
