We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by {it white Levy noise} in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Levy white noise parameters. Depending on the value of the skewness $beta$ of the Levy noise, the escape can either become enhanced or suppressed: a negative asymmetry $beta$ causes typically a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index $alpha$ which characterizes the jump length distribution of Levy noise, with a marked discontinuity occurring at $alpha=1$. We find that the typical factor of ``two that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior. Distinct non-exponential deviations occur, however, for low barrier heights.
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