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.
Graduate Course "Brownian motion"
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
Here is the list of the material I will be covering:
Peter Mörters and Yuval Peres.
Cambridge Series in Statistical and Probabilistic Mathematics, Volume 30.
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.
Homepage of Peter Mörters.