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A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter

Alexandre Richard 1, 2, 3
Abstract : Using structures of Abstract Wiener Spaces and their reproducing kernel Hilbert spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motion and multiparameter fractional Brownian motion, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.
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Alexandre Richard. A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter. Stochastic Processes and their Applications, Elsevier, 2015, 125 (4), pp.1394-1425. ⟨10.1016/j.spa.2014.11.003⟩. ⟨hal-00922028⟩

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