Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (J. Theor. Probab. 19(2):337-364, for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to satisfy a strengthened definition of increment stationarity. This new definition for stationarity property allows us 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.