Abstract
A continuous function x on the unit interval is an algorithmically random Brownian motion when every probabilistic event A which holds almost surely with respect to the Wiener measure, is reflected in x, provided A has a suitably effective description. In this paper we study the zero sets and global maxima from the left as well as the images of compact sets of reals of Hausdorff dimension zero under such a Brownian motion. In this way we shall be able to find arithmetical definitions of perfect sets of reals whose elements are linearly independent over the field of recursive real numbers.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Asarin, E.A., Prokovskiy, A.V.: Use of the Kolmogorov complexity in analyzing control system dynamics. Automat. Remote Control 47, 21–28 (1986); Translated from: Primeenenie kolmogorovskoi slozhnosti k anlizu dinamiki upravlemykh sistem. Automatika i Telemekhanika (Automation Remote Control) 1, 25–33 (1986)
Chaitin, G.A.: Algorithmic information theory. Cambridge University Press, Cambridge (1987)
Fouché, W.L.: Arithmetical representations of Brownian motion I. J. Symb. Logic 65, 421–442 (2000)
Fouché, W.L.: The descriptive complexity of Brownian motion. Advances in Mathematics 155, 317–343 (2000)
Fouché, W.L.: Dynamics of a generic Brownian motion: Recursive aspects, in: From Gödel to Einstein: Computability between Logic and Physics. Theoretical Computer Science 394, 175–186 (2008)
Freedman, D.: Brownian motion and diffusion, 2nd edn. Springer, New York (1983)
Hinman, P.G.: Recursion-theoretic hierarchies. Springer, New York (1978)
Kahane, J.-P.: Some random series of functions, 2nd edn. Cambridge University Press, Cambridge (1993)
Kjos-Hanssen, B., Nerode, A.: The law of the iterated logarithm for algorithmically random Brownian motion. In: Artemov, S., Nerode, A. (eds.) LFCS 2007. LNCS, vol. 4514, pp. 310–317. Springer, Heidelberg (2007)
Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)
Peres, Y.: An Invitation to Sample Paths of Brownian Motion, http://www.stat.berkeley.edu/~peres/bmall.pdf
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fouché, W.L. (2009). Fractals Generated by Algorithmically Random Brownian Motion. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-03073-4_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03072-7
Online ISBN: 978-3-642-03073-4
eBook Packages: Computer ScienceComputer Science (R0)