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Spherical and hyperbolic fractional Brownian motion. (English) Zbl 1112.60029

Summary: We define a fractional Brownian motion indexed by a sphere, or more generally by a compact rank one symmetric space, and prove that it exists if, and only if, \(0<H\leq 1/2\). We then prove that fractional Brownian motion indexed by a hyperbolic space exists if, and only if, \(0<H\leq 1/2\). At last, we prove that fractional Brownian motion indexed by a real tree exists when \(0<H\leq 1/2\).

MSC:

60G15 Gaussian processes
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