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

Abstract

The distribution of the first-passage time $T_a$ 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 $\mu \geq 0$. 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 $\mu <0$. 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