  The goal of the paper is to analytically examine escape probabilities for
dynamical systems driven by symmetric $\alpha$-stable L\'evy motions. Since
escape probabilities are solutions of a type of integro-differential equations
(i.e., differential equations with nonlocal interactions), asymptotic methods
are offered to solve these equations to obtain escape probabilities when noises
are sufficiently small. Three examples are presented to illustrate the
asymptotic methods, and asymptotic escape probability is compared with
numerical simulations.
