Unavoidable collections of balls for isotropic Lévy processes

A collection { B ¯ ( x n , r n ) } n ⩾ 1 of pairwise disjoint balls in the Euclidean space R d is said to be avoidable with respect to a transient process X if the process with positive probability escapes to infinity without hitting any ball. In this paper we study sufficient and necessary conditions for avoidability with respect to unimodal isotropic Lévy processes satisfying a certain scaling hypothesis. These conditions are expressed in terms of the characteristic exponent of the process, or alternatively, in terms of the corresponding Green function. We also discuss the same problem for a random collection of balls. The results are generalization of several recent results for the case of Brownian motion.