  The escape probability is a deterministic concept that quantifies some
aspects of stochastic dynamics. This issue has been investigated previously for
dynamical systems driven by Gaussian Brownian motions. The present work
considers escape probabilities for dynamical systems driven by non-Gaussian
L\'evy motions, especially symmetric $\alpha$-stable L\'evy motions. The escape
probabilities are characterized as solutions of the
  Balayage-Dirichlet problems of certain partial differential-integral
equations. Differences between escape probabilities for dynamical systems
driven by Gaussian and non-Gaussian noises are highlighted. In certain special
cases, analytic results for escape probabilities are given.
