On sample path properties of semistable processes

This paper contains three main results: In the first result a correspondence principle between semistable measures on Lp, 1 <= p < [infinity], and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to Lp with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.