Brownian Motion

Abstract

In this chapter, we construct Brownian motion and investigate some of its properties. We start by introducing “pre-Brownian motion”, which is easily defined from a Gaussian white noise on $$\mathbb{R}_{+}$$whose intensity is Lebesgue measure. Going from pre-Brownian motion to Brownian motion requires the additional property of continuity of sample paths, which is derived here via the classical Kolmogorov lemma. The end of the chapter discusses several properties of Brownian sample paths, and establishes the strong Markov property, with its classical application to the reflection principle.