We study the motion of a particle embedded in a time independent periodic
potential with broken mirror symmetry and subjected to a L\'evy noise
possessing L\'evy stable probability law (L\'evy ratchet). We develop
analytical approach to the problem based on the asymptotic probabilistic method
of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A
{\bf39}, L237 (2006); Stoch. Proc. Appl. {\bf116}, 611 (2006)]. We derive
analytical expressions for the quantities characterizing the particle motion,
namely the splitting probabilities of first escape from a single well, the
transition probabilities and the particle current. A particular attention is
devoted to the interplay between the asymmetry of the ratchet potential and the
asymmetry (skewness) of the L\'evy noise. Intensive numerical simulations
demonstrate a good agreement with the analytical predictions for sufficiently
small intensities of the L\'evy noise driving the particle.Comment: 14 pages, 11 figures, 63 reference
