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.
"Recurrent Sets for Transient Levy Processes with Bounded Kernels." Ann. Probab. 13 (4) 1204 - 1218, November, 1985. https://doi.org/10.1214/aop/1176992805