  The distribution for the minimum of Brownian motion or the Cauchy process is
well-known using the reflection principle. Here we consider the problem of
finding the sample-by-sample minimum, which we call the online minimum search.
We consider the possibility of the golden search method, but we show
quantitatively that the bisection method is more efficient. In the bisection
method there is a hierarchical parameter, which tunes the depth to which each
sub-search is conducted, somewhat similarly to how a depth-first search works
to generate a topological ordering on nodes. Finally, we consider the
possibility of using harmonic measure, which is a novel idea that has so far
been unexplored.
