Academics

Probability 3: A course on Brownian Motion and Stochastic Calculus

Time:Wed 09:50-12:15

Venue:Zoom: 787 662 9899 Password: BIMSA

Speaker:Yuval Peres (Professor)

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.

DATESeptember 3, 2024
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