Brownian local times

The purpose of this paper is to present in a more or less self-contained way the chief facts about the local times t of one-dimensional Brownian motion due to P. Lévy, F. Knight, D. B. Ray, and Itô-McKean. The deepest part concerns the remarkable fact that for a class of stopping times m, such as passage times and independent exponential holding times, the local time t(m, x) is a diffusion relative to its spatial parameter x. The beautiful methods of D. Williams are employed here as being most in the manner of P. Lévy who began the whole thing. The intent is purely expository, and only the main features of the proofs are indicated. A familiarity with the most elementary facts about Brownian motion is assumed. The paper is dedicated to Norman Levinson with affection and respect.