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Intermediate-Level Crossings of a First-Passage Path

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.