First-passage time distribution of a Brownian motion: two unexpected journeys

The distribution of the first-passage time (FPT) 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)=a2πt3e−(a−μt)22t,t>0, which is normalized only if μ≥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 μ < 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.