Directed transport induced by spatially modulated Lévy flights

In this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative α-stable Lévy noise, $\alpha \in (0,2),$ in the limit of a noise amplitude $\epsilon \to 0.$ We compare the dynamics for Itô and Marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells. A detailed analysis of the noise-induced current is performed for the Seebeck ratchet with a weak multiplicative noise for $\alpha \in (0,2].$ (paper)