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On the computability of a construction of Brownian motion†

Published online by Cambridge University Press:  17 May 2013

GEORGE DAVIE  and
WILLEM L. FOUCHÉ
GEORGE DAVIE
Affiliation:
Department of Decision Sciences, School of Economic Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa Email: davieg@unisa.ac.za; fouchwl@gmail.com
WILLEM L. FOUCHÉ
Affiliation:
Department of Decision Sciences, School of Economic Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa Email: davieg@unisa.ac.za; fouchwl@gmail.com
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Abstract

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Issue 6 , December 2013 , pp. 1257 - 1265
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

†

The research in this paper was supported by the National Research Foundation (NRF) of South Africa and by the European Union grant agreement PIRSES-GA-2011-2011-294962 in Computable Analysis (COMPUTAL).

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