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: (i) is at X at time T, or (ii) first reaches X 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 << 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 << 1 or X-2/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.