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 α -stable Lévy process on R 2 with 1 < α ≤ 2 and L Y ( 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 L Y ( ⋅ ) 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. 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 const / Ψ ( 1 , x ) where Ψ is the characteristic exponent.