On the one-sided exit problem for stable processes in random scenery

We consider the one-sided exit problem for stable Levy process in random scenery, that is the asymptotic behaviour for T large of the probability P[sup(t is an element of[0,T]) Delta(t) <= 1] where Delta(t) = integral(R) L-t(x) dW(x). Here W = {W(x); x is an element of R} is a two-sided standard real Brownian motion and {L-t(x); x is an element of R; t >= 0} the local time of a stable Levy process with index alpha is an element of (1, 2], independent from the process W. Our result confirms some physicists prediction by Redner and Majumdar.