First passage times for subordinate Brownian motions

Let XtXt be a subordinate Brownian motion, and suppose that the Lévy measure of the underlying subordinator has a completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(τx>t) of first passage times τxτx through a barrier at x>0x>0, and its derivatives in tt. As a corollary, we examine the asymptotic behaviour of P(τx>t) and its tt-derivatives, either as t→∞t→∞ or x→0+x→0+.