Shifted small deviations and Chung LIL for symmetric alpha-stable processes

Let Xα be a symmetric α-stable Lévy process with α ∈ (1, 2). We consider small ball probabilities of the following type P {‖Xα − λ f‖ < r} as r → 0, and λr → 0 or λr = c , c > 0, where ‖ · ‖ is the sup-norm and f ∈ C[0, 1] that starts at 0. We obtain the exact rate of decrease for these probabilities including constants. Using these small ball estimates, we obtain a functional LIL for Xα with attracting functions that are continuous. It occurs that the limit set of the family {