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Oscillatory Fractional Brownian Motion

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Abstract

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. A randomness property of the initial condition increases the long range dependence. We emphasize the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.

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Acknowledgement

This work was supported in part by Conacyt grant 98998 (Mexico) and MNiSzW grant N N201 397537 (Poland).

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    T. Bojdecki & A. Talarczyk

  2. Department of Mathematics, Centro de Investigación y de Estudios Avanzados, Mexico City, Mexico

    L. G. Gorostiza

Authors
  1. T. Bojdecki
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  2. L. G. Gorostiza
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  3. A. Talarczyk
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Correspondence to L. G. Gorostiza.

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Bojdecki, T., Gorostiza, L.G. & Talarczyk, A. Oscillatory Fractional Brownian Motion. Acta Appl Math 127, 193–215 (2013). https://doi.org/10.1007/s10440-013-9798-3

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  • DOI: https://doi.org/10.1007/s10440-013-9798-3

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