Stable processes with drift on the line

The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets. 1. Statement of results Let be a process with stationary independent increments having log characteristic function -*|0|°(l-iAsgn(0)), where 0 0 . There is an enormous literature on the behavior of drift free stable processes. Surprisingly, little has been written on stable processes with drift. In view of the extensive results known for the potential theory [6] and the path behavior [3, 4] for infinitely divisible processes in general, the interest today in special processes such as stable processes with drift lies in the fact that for such processes rather explicit results may be obtained. The incorporation of a drift term changes the behavior of the process as compared to a drift free process. However the drift acts substantially different for processes with index a 1 . In all cases except for a 1 and s = 0 (i.e. the symmetric Cauchy process) the processes with drift are transient and hit points with positive probability. This fact follows from the general theory [3 and 6]. This is in contrast to the drift free processes. For drift free processes if a < 1 the processes are transient but do not hit points, Received by the editors July 31, 1986 and, in revised form, February 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 60J30; Secondary 60J45.