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We study the barrier crossing of a particle driven by white symmetric Levynoise of index $\alpha$ and intensity $DD for three different generic types ofpotentials: (a) a bistable potential; (b) a metastable potential; and (c) atruncated harmonic potential. For the low noise intensity regime we recover thepreviously proposed algebraic dependence on $D$ of the characteristic escapetime, $T_{\mathrm{esc}}\simeq C(\alpha)/D^{\mu(\alpha)}$, where $C(\alpha)$ isa coefficient. It is shown that the exponent $\mu(\alpha)$ remainsapproximately constant, $\mu\approx 1$ for $0<\alpha<2$; at $\alpha=2$ thepower-law form of $T_{\mathrm{esc}}$ changes into the known exponentialdependence on 1/D; it exhibits a divergence-like behavior as $\alpha$approaches 2. In this regime we observe a monotonous increase of the escapetime $T_{\mathrm{esc}}$ with increasing $\alpha$ (keeping the noise intensity$D$ constant). The probability density of the escape time decays exponentially.In addition, for low noise intensities the escape times correspond to barriercrossing by multiple Levy steps. For high noise intensities, the escape timecurves collapse for all values of $\alpha$. At intermediate noise intensities,the escape time exhibits non-monotonic dependence on the index $\alpha$, whilestill retaining the exponential form of the escape time density.

Barrier crossing driven by Lévy noise: Universality and the role of noise intensity