Path Properties of Brownian Motion

Chapter 3 investigates a variety of path-based properties of Brownian motion in the general categories of symmetry, growth, regularity (i.e., smoothness), and variation. To simplify the exposition, most results are stated in the context of a 1-dimensional Brownian motion, with generalizations noted. For symmetry, d-dimensional Brownian motion is seen to be rotationally invariant, while 1-dimensional Brownian transforms predictably with respect to time translation, scaling, reversal, and inversion. The investigation into growth begins with bounds on the tail probabilities of the supremum of | B t | over [0, T]. The strong law of large numbers is the first of several investigations of B t as t→∞, and this result is then refined to the sharp law of the iterated logarithm of Aleksandr Khinchin (1894–1959). Regularity results begin with an investigation into Hölder continuity of 1-dimensional Brownian, with a positive result for 0