  Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in (1,2)$.
We study shifted small ball probabilities for these processes in the uniform
topology, when the shift function is an arbitrary continuous function which
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
$\alpha$-stable L\'evy process with attracting functions that are continuous.
It occurs that the limit set for the family of renormalized $\alpha$-stable
L\'evy processes is equal to the set of all continuous functions on $[0,1]$
which start at 0, under certain choice of normalizing functions.
