The Special Role of Brownian Motion

Brownian motion plays a universally important role among the continuous martingales. This will be illustrated in the subsequent sections. The first one deals with what we view as the most important result on Brownian motion (besides Wiener’s existence theorem) namely P. Levy’s characterization by means of quadratic variation. The theorem of K. Dambis, L.E. Dubins and G. Schwarz in section 9.2 tells us that every a.s. continuous local martingale is a time-changed Brownian motion. The next two sections contain aspects of Brownian motion which make it a very special Markov process. Section 9.5 presents Levy’s theorem on conformai invariance of complex Brownian motion. Section 9.6 introduces Hermite polynomials and indicates their role in integration w.r.t. Brownian motion and section 9.7 shows that every martingale for the natural filtration of a Brownian motion automatically is a.s. continuous.