Recurrent lines in two-parameter isotropic stable Lévy sheets

It is well known that an Rd-valued isotropic α-stable Lévy process is (neighborhood-)recurrent if and only if d⩽α. Given an Rd-valued two-parameter isotropic α-stable Lévy sheet {X(s,t)}s,t⩾0, this is equivalent to saying that for any fixed s∈[1,2], P{t↦X(s,t) is recurrent}=0 if d>α and =1 otherwise. We prove here that P{∃s∈[1,2]: t↦X(s,t) is recurrent}=0 if d>2α and =1 otherwise. Moreover, for d∈(α,2α], the collection of all times s at which t↦X(s,t) is recurrent is a random set of Hausdorff dimension 2−d/α that is dense in R+, a.s. When α=2, X is the two-parameter Brownian sheet, and our results extend those of Fukushima and Kôno.