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A Set-indexed Fractional Brownian Motion

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We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.

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Authors and Affiliations

  1. Dassault Aviation, 78 quai Marcel Dassault, 92552, Saint-Cloud Cedex, France

    Erick Herbin

  2. Dept. of Mathematics, Bar Ilan University, 52900, Ramat-Gan, Israel

    Ely Merzbach

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  1. Erick Herbin
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  2. Ely Merzbach
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Correspondence to Ely Merzbach.

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Herbin, E., Merzbach, E. A Set-indexed Fractional Brownian Motion. J Theor Probab 19, 337–364 (2006). https://doi.org/10.1007/s10959-006-0019-0

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  • DOI: https://doi.org/10.1007/s10959-006-0019-0

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