The distribution of the first-passage time Ta​ for a Brownian particle with
drift μ subject to hitting an absorber at a level a>0 is well-known and
given by its density γ(t)=2πt3​a​e−2t(a−μt)2​,t>0, which is normalized only if μ≥0. In this article,
we show that there are two other families of diffusion processes (the first
with one parameter and the second with two parameters) having the same first
passage-time distribution when μ<0. In both cases we establish the
propagators and study in detail these new processes. An immediate consequence
is that the distribution of the first-passage time does not allow us to know if
the process comes from a drifted Brownian motion or from one of these new
processes.Comment: 16 pages, 2 figures; improved mathematical notatio