Brownian Motion

Abstract

We begin this chapter with the definition of Brownian motion and a proof that its distribution is supported by the space of continuous functions (§2.1), and then go on to deal with important aspects of Brownian motion such as its sample path (§2.2) and Markov properties (§2.4). It is through these discussions that we can appreciate the place of Brownian motion within the class of all stochastic processes and, in particular, Gaussian processes. Two methods of constructing Brownian motion will be presented (§2.3), each of which is significant in its own right, and which also exhibits the ideas underlying constructions relevant to later chapters. Markov properties will only be touched upon briefly (§§2.4–2.6), but, hopefully, enough to enable a close connection with analysis to be seen.