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.

翻译：利用反射原理，布朗运动或柯西过程的最小值分布已是众所周知。本文考虑逐样本最小值问题，将其称为在线最小值搜索。我们探讨了黄金搜索法的可行性，但定量证明二分法更为高效。在二分法中存在层次化参数，用于调节各子搜索的深度，其方式类似于深度优先搜索生成节点拓扑排序。最后，我们考虑使用调和测度的可能性，这是一个迄今尚未被探索的新颖思路。