Before revision:   R_+ \to[-\infty,+\infty] such that the survival function of the first-passage time \tau_b=\inf\{t\ge0: X(t) \leq b(t)\} is given by  \overline H. In this paper we consider a version of the IFPT problem where the barrier is  fixed at zero and the problem is to find an entrance law \mu and a time-change I such that for the time-changed process X\circ I the IFPT problem is solved by a constant barrier at the level zero.
After revision:   \mathbb R^+ \to[-\infty,+\infty] such that the survival function of the first-passage time \tau_b=\inf\{t\ge0: X(t) \leq b(t)\} is given by  \overline H. In this paper we consider a version of the IFPT problem where the barrier is  fixed at zero and the problem is to find an entrance law \mu and a time-change I such that for the time-changed process X\circ I the IFPT problem is solved by a constant barrier at the level zero.