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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
