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

Abstract

We study the barrier crossing of a particle driven by white symmetric L\'evy noise of index $\ensuremath{\alpha}$ and intensity $D$ for three different generic types of potentials: (a) a bistable potential, (b) a metastable potential, and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on $D$ of the characteristic escape time, ${T}_{\mathrm{esc}}\ensuremath{\simeq}C(\ensuremath{\alpha})∕{D}^{\ensuremath{\mu}(\ensuremath{\alpha})}$, where $C(\ensuremath{\alpha})$ is a coefficient. It is shown that the exponent $\ensuremath{\mu}(\ensuremath{\alpha})$ remains approximately constant, $\ensuremath{\mu}\ensuremath{\approx}1$ for $0<\ensuremath{\alpha}<2$; at $\ensuremath{\alpha}=2$ the power-law form of ${T}_{\mathrm{esc}}$ changes into the known exponential dependence on $1∕D$; it exhibits a divergencelike behavior as $\ensuremath{\alpha}$ approaches 2. In this regime we observe a monotonous increase of the escape time ${T}_{\mathrm{esc}}$ with increasing $\ensuremath{\alpha}$ (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 barrier crossing by multiple L\'evy steps. For high noise intensities, the escape time curves collapse for all values of $\ensuremath{\alpha}$. At intermediate noise intensities, the escape time exhibits nonmonotonic dependence on the index $\ensuremath{\alpha}$, while still retaining the exponential form of the escape time density.