Small deviations of weighted fractional processes and average non–linear approximation

Abstract

We investigate the small deviation problem for weighted fractional Brownian motions in L q L_q –norm, 1 ≤ q ≤ ∞ 1\le q\le \infty . Let B H B^H be a fractional Brownian motion with Hurst index 0 > H > 1 0>H>1 . If 1 / r := H + 1 / q 1/r:=H+1/q , then our main result asserts \[ lim ε → 0 ε 1 / H log ⁡ P ( ‖ ρ B H ‖ L q ( 0 , ∞ ) > ε ) = − c ( H , q ) ⋅ ‖ ρ ‖ L r ( 0 , ∞ ) 1 / H , \lim _{\varepsilon \to 0} \varepsilon ^{1/H}\log \mathbb {P}\left ({\left \|{\rho \,B^H}\right \|_{L_q(0,\infty )}>\varepsilon } \right ) = -c(H,q)\cdot \left \|{\rho }\right \|_{L_r(0,\infty )}^{1/H}, \] provided the weight function ρ \rho satisfies a condition slightly stronger than the r r –integrability. Thus we extend earlier results for Brownian motion, i.e. H = 1 / 2 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 l q l_q –sums of linear operators defined on a Hilbert space.