Journal of Theoretical Probability

, Volume 19, Issue 2, pp 337–364

A Set-indexed Fractional Brownian Motion


DOI: 10.1007/s10959-006-0019-0

Cite this article as:
Herbin, E. & Merzbach, E. J Theor Probab (2006) 19: 337. doi:10.1007/s10959-006-0019-0

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.


Fractional Brownian motionGaussian processesstationarityself-similarityset-indexed processes

AMS Classification


Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Dassault AviationSaint-Cloud CedexFrance
  2. 2.Dept. of MathematicsBar Ilan UniversityRamat-GanIsrael