Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion

Erick Herbin; Ely Merzbach

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arXiv · 0706.3472

Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion

Erick Herbin·Ely Merzbach

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Abstract

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.

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Erick Herbin, Ely Merzbach. 2008-07-09. Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion. https://arxiv.org/abs/0706.3472

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