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

TL;DR

This paper reveals that different diffusion processes can share identical first-passage time distributions, challenging the assumption that such distributions uniquely identify the underlying process, especially when the drift is negative.

Contribution

It introduces two new classes of diffusion processes with the same FPT distribution as drifted Brownian motion for negative drift, and characterizes their transition densities.

Findings

Two new process categories share FPT distribution with drifted Brownian motion for negative drift.

Transition densities of the new processes are explicitly identified.

FPT distribution alone cannot distinguish between the original and new processes.

Abstract

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.