2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164389The 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<1.18 pagesProbability60G10; 60G15; 60G17; 60G18text