  The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle
with drift $\mu$ subject to hitting an absorber at a level $a>0$ is well-known
and given by its density $\gamma(t) = \frac{a}{\sqrt{2 \pi t^3} }
e^{-\frac{(a-\mu t)^2}{2 t}}, t>0$, which is normalized only if $\mu \geq 0$.
This article demonstrates the existence of two additional diffusion process
categories (one with one parameter and the other with two) that have the same
first passage-time distributions when $\mu <0$. For both, we identify the
transition densities and thoroughly investigate the processes. A substantial
implication is that the first-passage time distribution does not indicate
whether the process originates from a drifted Brownian motion or from one of
the new processes presented.
