Recurrent Sets for Transient Levy Processes with Bounded Kernels

Abstract

In the study of recurrent sets for transient Levy processes on the real line, we present two main results. As long as the process has a "well-behaved" (bounded in a particular way) kernel, a set is recurrent for the process if and only if the sum of the capacities of pieces of the set is infinite. In the second result, we show that a simple condition on the Levy measure guarantees that the process has a "well-behaved" kernel. Finally, the results are applied to subordinators in order to construct examples of recurrent sets including a recurrent set with finite Lebesgue measure.