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 the 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/DT 1. We also determine the time 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/DT 1 or X/DT 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. PACS numbers: 05.40.Jc, 02.50.-r, 05.40.-a Intermediate-Level Crossings of a First-Passage Path 2