Hitting of a line or a half-line in the plane by two-dimensional symmetric stable Lévy processes

Let (X(t),Y(t))(X(t),Y(t)) be a symmetric αα-stable Lévy process on R2R2 with 1<α≤21<α≤2 and LY(t)LY(t) be the local time at 00 for Y(t)Y(t). A multivariate asymptotic estimate is obtained involving the first hitting time and place of the positive half of the XX-axis, and LY(⋅)LY(⋅) up to then. The method is based on the fluctuation identities for two-dimensional processes and the same method is applicable for a wider class of processes.When (X(0),Y(0))=(0,1)(X(0),Y(0))=(0,1), the law of the first hitting place of the whole XX-axis is shown to have the explicit density const/Ψ(1,x) where ΨΨ is the characteristic exponent.