Ballistic Lévy walk with rests: Escape from a bounded domain

Abstract

The Lévy walk process for the lower interval of the time of flight distribution ($\alpha <1$) and with finite resting time between consecutive flights is discussed. The motion is restricted to a region bounded by two absorbing barriers and the escape process is analysed. By means of a Poisson equation, the total density, which includes both flying and resting phase, is derived and the first passage time properties determined: the mean first passage time appears proportional to the barrier position; moreover, the dependence of that quantity on $\alpha$ is established. Two limits emerge from the model: of short waiting time, that corresponds to Lévy walks without rests, and long waiting time which exhibits properties of a Lévy flights model. The similar quantities are derived for the case of a position-dependent waiting time. Then the mean first passage time rises with barrier position faster than for Lévy flights model. The analytical results are compared with Monte Carlo trajectory simulations.

8 pages, 5 figures

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