Chapter 7 - One-Dimensional Diffusions and Their Convergence in Distribution

Abstract

This chapter examines the one-dimensional diffusions and their convergence in distribution. The normalized Brownian motion is analyzed in the chapter. To obtain one-dimensional Brownian motion, fix an axis and project the motion of the particle on this axis. The Brownian traveler starts afresh with respect to path shifts induced by random times. One of the best ways to appreciate the wilderness that is filled with exotic forms is to delve into the study of the sample path properties of Brownian motion. The universal nature of Brownian motion is revealed in the sense that this process can be used to capture random walks. The scaled random walk derived from fair coin tossing converges weakly to normalized Brownian motion. This not only gives insight into Brownian motion, but it displays the power of weak convergence. Diffusion in natural scale has the same state space features as Brownian motion. The problem of describing the speed of diffusion in natural scale in terms of its speed measure is elaborated in the chapter. The chapter also analyzes the local time for Brownian motion.