Transport in a Lévy ratchet: group velocity and distribution spread

We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity.