Constructions of Brownian Motion

Chapter 2 develops several implementable constructions of a 1-dimensional Brownian motion defined on [0, ∞) or [0, 1] on variously defined probability spaces (S,σ(S),μ). By Chapter 1 , such constructions are sufficient to provide a construction of d-dimensional Brownian motion on this same time interval. In contrast to the Kolmogorov results of the Chapter 1 which focus on existence, the four approaches in this chapter are genuinely constructive. They can be implemented with a computer program, and will produce Brownian-like paths to any degree of refinement desired. In addition, all use interesting and useful mathematics. The first approach by Monroe D. Donsker (1924–1991) introduces random walks, and provides a significant generalization of the Chapter VI.10 investigation into distributional binomial motion. The next three approaches, by Paul Lévy (1886–1971), Zbigniew Ciesielski (1934–2020), and Norbert Wiener (1894–1964), provide series-based convergence results reminiscent of the path-based binomial model of Book VI, but with normal variates and various interpolating functions.