A fractional Brownian field indexed by L2 and a varying Hurst parameter

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0, 1/2] × L2(T, m), (T, m) a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brow-nian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satis-factory 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