Recurrent lines in two-parameter isotropic stable Lévy sheets

It is well known that an -valued isotropic [alpha]-stable Lévy process is (neighborhood-)recurrent if and only if d[less-than-or-equals, slant][alpha]. Given an -valued two-parameter isotropic [alpha]-stable Lévy sheet {X(s,t)}s,t[greater-or-equal, slanted]0, this is equivalent to saying that for any fixed s[set membership, variant][1,2], P{t|->X(s,t) is recurrent}=0 if d>[alpha] and =1 otherwise. We prove here that P{[there exists]s[set membership, variant][1,2]: t|->X(s,t) is recurrent}=0 if d>2[alpha] and =1 otherwise. Moreover, for d[set membership, variant]([alpha],2[alpha]], the collection of all times s at which t|->X(s,t) is recurrent is a random set of Hausdorff dimension 2-d/[alpha] that is dense in , a.s. When [alpha]=2, X is the two-parameter Brownian sheet, and our results extend those of Fukushima and Kôno.