Small Deviations of Weighted Fractional Processes and Average Non–linear Approximation

We investigate the small deviation problem for weighted fractional Brownian motions in Lq–norm, 1 ≤ q ≤ ∞. Let BH be a fractional Brownian motion with Hurst index 0 < H < 1. If 1/r := H + 1/q, then our main result asserts lim ε→0 ε log P (∥∥∥ρBH∥∥∥ Lq(0,∞) < ε ) = −c(H, q) · ‖ρ‖ Lr(0,∞) , provided the weight function ρ satisfies a condition slightly stronger than the r– integrability. Thus we extend earlier results for Brownian motion, i.e. H = 1/2, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non–linear approximation technique for Gaussian processes as well as sharp entropy estimates for lq–sums of linear operators defined on a Hilbert space.