✦ LIBER ✦

The Descriptive Complexity of Brownian Motion

✍ Scribed by Willem L. Fouché

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 205 KB
Volume: 155
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.2000.1945

No coin nor oath required. For personal study only.

✦ Synopsis

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.

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