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Using structures of Abstract Wiener Spaces, we define a fractional Brownianfield indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separablemeasure space, where the first coordinate corresponds to the Hurst parameter offractional Brownian motion. This field encompasses a large class of existingfractional Brownian processes, such as L\'evy fractional Brownian motions andmultiparameter fractional Brownian motions, and provides a setup for new ones.We prove that it has satisfactory incremental variance in both coordinates andderive certain continuity and H\"older regularity properties in relation withmetric entropy. Also, a sharp estimate of the small ball probabilities isprovided, generalizing a result on L\'evy fractional Brownian motion. Then, weapply these general results to multiparameter and set-indexed processes,proving the existence of processes with prescribed local H\"older regularity ongeneral indexing collections.

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