Intermediate-Level Crossings of a First-Passage Path

TL;DR

This paper analyzes the properties of one-dimensional Brownian paths, focusing on their most likely trajectories, crossing times, and how these depend on the relation between spatial and temporal scales, revealing surprising behaviors.

Contribution

It provides new analytical results on the most probable paths and crossing time distributions of Brownian motion, including regimes with unimodal and bimodal crossing time distributions.

Findings

Most likely paths are repelled from the final point when $X^2/DT\ll 1$

Distribution of first-crossing times can be unimodal or bimodal depending on $X^2/DT$

First-crossing probability in the bimodal regime resembles a more singular arcsine law

Abstract

We investigate some simple and surprising properties of a one-dimensional Brownian trajectory with diffusion coefficient $D$ that starts at the origin and reaches $X$ either: (i) at time $T$ or (ii) for the first time at time $T$. We determine the most likely location of the first-passage trajectory from $(0,0)$ to $(X,T)$ and its distribution at any intermediate time $t<T$. A first-passage path typically starts out by being repelled from its final location when $X^2/DT\ll 1$. We also determine the distribution of times when the trajectory first crosses and last crosses an arbitrary intermediate position $x<X$. The distribution of first-crossing times may be unimodal or bimodal, depending on whether $X^2/DT\ll 1$ or $X^2/DT\gg 1$. The form of the first-crossing probability in the bimodal regime is qualitatively similar to, but more singular than, the well-known arcsine law.