Abstract
The English botanist Brown observed in 1826 that microscopic particles suspended in a liquid are subject to continual molecular impacts and execute zigzag movements—Brownian motion. Einstein found that these movements can be analyzed by laws of probability. One of the simplest models for a one dimensional Brownian motion can be given in terms of the coin tossing or random walk model. Various lemmas and theorems are proved in the chapter. The law of iterated logarithm for the Wiener process, and the Brownian bridges are reviewed. The chapter also reviews the distributions of some functional of the Wiener and Brownian bridge processes, and the modulus of non-differentiability of the Wiener process. Almost all sample functions of a Wiener process are nowhere differentiable. The chapter discusses the infinite series representations of the Wiener process and Brownian bridge, and reviews the Ornstein-Uhlenbeck process. The Kiefer process is also discussed.
Published Version
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