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Many physical and chemical phenomena are governed by stochastic escape acrosspotential barriers. The escape time depends on the structure of the noise andthe shape of the potential barrier. By applying $\alpha$-stable noise from the$\alpha=2$ Gaussian noise limit to the $\alpha<2$ jump processes, we find acontinuous transition of the mean escape time from the usual dependence on theheight of the barrier for Gaussian noise to a dependence solely on the width ofthe barrier for $\alpha$-stable noise. We consider the exit problem of aprocess driven by $\alpha$-stable noise in a double well potential. We studyindividually the influences of the width and the height of the potentialbarrier in the escape time and we show through scalings that the asymptoticlaws are described by a universal curve independent of both parameters. Whenthe dependence in the stability parameter is considered, we see that there aretwo different diffusive regimes in which diffusion is described either byKramer's time or by the corresponding asymptotic law for $\alpha$-stable noise.We determine the regions of the noise parameter space in which each regimeprevails, and exploit this result to construct an anomalous example in which adouble well potential exhibit a different diffusion regime in each well for awide range of parameters.

Escape by jumps and diffusion by {\alpha}-stable noise across the barrier in a double well potential