The Descriptive Complexity of Brownian Motion

Abstract

A continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x, provided that the event has a suitably effective description. We show that a generic one-dimensional Brownian motion can be computed from an infinite binary string which is complex in the sense of Kolmogorov–Chaitin. Conversely, one can construct a Kolmogorov–Chaitin random string from the values at the rational numbers of a generic Brownian motion. In this way, we construct a recursive isomorphism between encoded versions of generic Brownian motions and Kolmogorov–Chaitin random reals.