Probability Graduate Students Seminar Introduction to the theory of Brownian Local Times

Local Times of Brownian Motion were introduced in 1948 by French mathematician Paul Lévy in his fundamental book Processus Stochastiques et Mouvement Brownien. Naturally arising in many problems (such as, for instance, an extension of Itô’s formula to convex functions, or finding the density of the occupation measure of BM with respect to the Lebesgue measure), they also roughly describe the amount of time spent by a standard real brownian motion near a given point, thus providing a very fine description of the sample paths of BM. We will start by giving several constructions of Brownian local times and their basic properties. Then we will illustrate both the power and the elegance of the theory by proving several theorems of great importance in the study of Brownian motion and real stochastic processes. These include : P. Lévy’s arcsine law, the Lévy identity, and the beautiful theorems of Ray and Knight on the spatial behaviour of local times.