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The mean first exit time and escape probability are utilized to quantifydynamical behaviors of stochastic differential equations with non-Gaussianalpha-stable type Levy motions. Both deterministic quantities are characterizedby differential-integral equations(i.e.,differential equations with non localterms) but with different exterior conditions. The non-Gaussianity of noisesmanifests as nonlocality at the level of mean exit time and escape probability.An objective of this paper is to make mean exit time and escape probability asefficient computational tools, to the applied probability community, forquantifying stochastic dynamics. An accurate numerical scheme is developed andvalidated for computing the mean exit time and escape probability. Asymptoticsolution for the mean exit time is given when the pure jump measure in the Levymotion is small.  From both the analytical and numerical results, it is observed that the meanexit time depends strongly on the domain size and the value of alpha in thealpha-stable Levy jump measure. The mean exit time can measure which of the twocompeting factors in alpha-stable Levy motion, i.e. the jump frequency or thejump size, is dominant in helping a process exit a bounded domain. The escapeprobability is shown to vary with the underlying vector field(i.e.,drift). Themean exit time and escape probability could become discontinuous at theboundary of the domain, when the process is subject to certain deterministicpotential and the value of alpha is in (0,1).

Mean exit time and escape probability for dynamical systems driven by Levy noise