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 spaces $L_q([0, 1], \mu)$. Of special interest here are thin (fractal) measures $\mu$, that is, those that are singular with respect to the Lebesgue measure. We describe the behavior of small deviation probabilities by numerical quantities of $\mu$, called mixed entropy numbers, characterizing size and regularity of the underlying measure. For the particularly interesting case of self-similar measures, the asymptotic behavior of the mixed entropy is evaluated explicitly. We also provide two-sided estimates for this quantity in the case of random measures generated by subordinators.

While the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods, the lower bound is verified by analytic tools concerning entropy and Kolmogorov numbers of Riemann--Liouville operators.