Brownian Motion, 2nd Edition

3 Constructions of Brownian motion

There are several different ways to construct Brownian motion and we will present a few of them here. Since a d-dimensional Brownian motion can be obtained from d independent one-dimensional Brownian motions, see Section 2.2, we restrict our attention to the one-dimensional setting. If this is your first encounter with Brownian motion, you should restrict your attention to Sections 3.1 and 3.2 or Section 3.4 which contain the most intuitive constructions.

3.1 A random orthogonal series

Some of the earliest mathematical constructions of Brownian motion are based on random orthogonal series. The idea is to write the paths [0,1] ∋ t ↦ B_t(ω) for (almost) every ω as a random series with respect to a complete orthonormal ...