Much of probability theory is devoted to describing the macroscopic picture emerging in random systems defined by a host of microscopic random effects. Brownian motion is the macroscopic picture emerging from a particle moving randomly on the real line. On the microscopic level, at any time step, the particle receives a random displacement, caused for example by other particles hitting it or by an external force, so that its position at time n is given as sum of n displacements, which are assumed to be independent, identically distributed random variables. The resulting process is a random walk, the displacements represent the microscopic inputs. It turns out that not all the features of the microscopic inputs contribute to the macroscopic picture. Indeed, if they exist, only the mean and covariance of the displacements are shaping the picture. In particular, all random walks whose displacements have the same mean and covariance give rise to the same macroscopic process, and even the assumption that the displacements have to be independent and identically distributed can be substantially relaxed. This effect is called universality, and the macroscopic process is often called a universal object. It is a common approach in probability to study various phenomena through the associated universal objects.

The aim of this course is to introduce Brownian motion as the universal object describing macroscopically the continuous movements of a driftless particle on the real line. We shall discuss its properties, putting particular emphasis on the sample path properties. The course is based on the recent graduate text book, which you can download here

• Brownian motion
Peter Mörters and Yuval Peres.
Cambridge Series in Statistical and Probabilistic Mathematics, Volume 30.

Here is the list of the material I will be covering:
• Lecture 1: Levy's construction of Brownian motion and modulus of continuity.
• Lecture 2: Paley-Wiener-Zygmund theorem on non-differentiability of paths, Markov property and Blumenthal's zero-one law.
• Lecture 3: The strong Markov property, reflection principle and applications.
• Lecture 4: The martingale property and applications, in particular hitting times and probabilities.
• Lecture 5: Hausdorff dimension of the zero-set of Brownian motion.
• Lecture 6: Embedding random walks in Brownian motion, and the Donsker invariance principle.
• Lecture 7: Application: Arc sine laws.
• Lecture 8: Application: Pitman's 2M-X theorem.
• Lecture 9: Local time of Brownian motion.
• Lecture 10: The Ray-Knight theorem.

Prerequisites for the course are an undergraduate course in probability of about third year level and good personal motivation.

Students on this course requiring credit can download the take-home worksheet. Please answer exactly three of the six questions and send them to

Peter Mörters, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

by January 13th 2012 at the latest. Please enclose a signed note confirming that you have done the work yourself, and stick to the rules on plagiarism in your university.

The course is part of the EPSRC funded Mathematics Taught Course Centre.

Last changed on 30.11.2011.
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