Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures

We investigate Riemann–Liouville processes R_H, with H> 0, and fractional Brownian motions B_H, for 0 < H <1, and study their small deviation properties in the spacesL_q(0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of Riemann–Liouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary).