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Fractional Brownian motion run with a nonlinear clock

Daniel O’Malley*

John H. Cushman†

  • Department of Mathematics, Purdue University, West Lafayette, Indiana 47097, USA

  • Department of Earth and Atmospheric Sciences and Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

  • *omalled@math.purdue.edu
  • †jcushman@purdue.edu
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Phys. Rev. E 82, 032102 – Published 20 September, 2010

DOI: https://doi.org/10.1103/PhysRevE.82.032102

Abstract

We construct a family of stochastic processes with nonstationary, correlated increments which allow a priori independent selections of both fractal dimension and mean-square displacement. The family is essentially fractional Brownian motion (fBm) run with a nonlinear clock (fBm-nlc). The fractal dimension of fBm-nlc is shown to be the same as that of the underlying fBm process. We also compute the p-variation and discuss the problems in using this to differentiate between diffusive processes. The fBm-nlc process illustrates that the range of anomalous diffusive processes has not been adequately explored.

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Outline

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Phys. Rev. E 82, 032102– Published 20 September, 2010
  • Received 19 February 2010

© 2010 American Physical Society

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