Abstract. Let $X(t) (t \in{\msbm R}_+)$ be a stable process in a random scenery. The Hausdorff dimension of certain level sets is determined and the existence of the local time of $X(t)$ is proved.

AMS Subject Classification (1991): 60G17, 60G18

Received August 4, 1998 and in revised form January 4, 1999. (Registered under 2692/2009.)