Brownian Motion

This is a set of lecture notes based on a graduate course given at the Taught Course Centre in Mathematics in 2011. The course is based on a selection of material from my book with Yuval Peres, entitled Brownian motion, which was published by Cambridge University Press in 2010. 1 Lévy's construction of Brownian motion and modulus of continuity 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 a line without making very big jumps. 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, if its position at time zero is S 0 , its position at time n is given as S n = S 0 + n i=1 X i , where the displacements X 1 , X 2 , X 3 ,. .. are assumed to be independent, identically distributed random variables with mean zero. The process {S n : n 0} 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, all random walks whose displacements have zero mean and variance one 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. If the jumps of a random walk are sufficiently tame to become negligible in the macroscopic picture, any continuous time stochastic process {B(t) : t 0} describing the macroscopic features of this random walk should have the following properties: (i) for all times 0 t 1 t 2. .. t n the random variables B(t n) − B(t n−1), B(t n−1) − B(t n−2),. .. , B(t 2) − B(t 1) are independent; (ii) the distribution of the increment B(t + h) − B(t) has zero mean and does not depend on t; (iii) the process {B(t) : t 0} has almost surely continuous paths.