A set-indexed fractional Brownian motion

Details A set-indexed fractional Brownian motion A set-indexed fractional Brownian motion

2005-03-11

arxiv.org/abs/math/0503211v3

Mathematics

Probability

24 pages

Scientific paper

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 Levy 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.

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