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

Let (X(t),Y(t)) be a symmetric [alpha]-stable Lévy process on with 1<[alpha]<=2 and LY(t) be the local time at 0 for Y(t). A multivariate asymptotic estimate is obtained involving the first hitting time and place of the positive half of the X-axis, and LY([dot operator]) 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), the law of the first hitting place of the whole X-axis is shown to have the explicit density where [Psi] is the characteristic exponent.