Journal of Theoretical Probability

, Volume 19, Issue 2, pp 337–364

A Set-indexed Fractional Brownian Motion

Authors

  • Erick Herbin
    • Dassault Aviation
    • Dept. of MathematicsBar Ilan University
Article

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.

Keywords

Fractional Brownian motion Gaussian processes stationarity self-similarity set-indexed processes

AMS Classification

62G05 60G15 60G17 60G18

Copyright information

© Springer Science+Business Media, Inc. 2006