Prerequisites
Students are required to have followed a basic (undergraduate) probability theory course, as well as to have a very basic knowledge of measure theory (any undergraduate course would suffice)
If a student completed the Measure Theoretic Probability or Stochastic Processes course within MasterMath, that would definitely be enough.
If knowledge of basic concepts of measure theory is a problem, potential students are advise to consult a concise set of notes, which can be found at http://www.math.rug.nl/~valesin/review_probability.pdf
Aim of the course
In this course we will study a particular class of Markov chains called random walks. One can visualize a random walker as a particle on the lattice Z^d, jumping from any position to one of the neighboring positions at random. Why is this particular class of random processes so important and why does it deserve a dedicated course?
The answer lies in the fact that random walks are ubiquitous in modern probability theory, and play a key role in the study of most classes of stochastic processes for which a theory has been developed. For instance, random walks are intimately related to Brownian motion, and hence to more general diffusion processes. In many contexts, random walks are useful in the understanding of more sophisticated probabilistic models. A solid grasp of random walks is thus part of the repertoire of mathematicians who deal with randomness in any way.
The theory of random walks is very well developed -- it has its beginnings together with probability theory itself, in letters between Fermat and Pascal discussing the gambler's ruin problem. This theory contains many very deep, and sometimes unexpected results. Moreover, it has deep links to other areas of mathematics such as harmonic analysis, differential equations and even complex analysis.
The course will start by treating random walks on the d-dimensional lattice Z^d, in the spirit of references [1] and [2] in the literature. Special emphasis will be given to nearest-neighbor and finite range walks on Z^d, though more general increment distributions will also be touched upon. We will begin with the classical theory, including gambler's ruin estimates and classification of random walks (periodicity, recurrence). We then cover more specialized topics such as the local central limit theorem (using Fourier analysis and focusing on error estimates) and Green function estimates. We will also discuss coupling methods and related asymptotics.
Beyond Z^d, we will also cover random walks on other graphs, following reference [3] in the literature, focusing on the well-known and very fruitful analogy with electrical networks. Time permitting, more advanced topics will be covered, such as: cover times and random walks on random environments.
This material will expose participants to a number of important tools of probability theory, such as couplings, martingales, stochastic domination and the second moment method.
Lecturers
Daniel Valesin, Evgeny Verbitskiy
- Docent: Evgeny Verbitskiy