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有成熟结论。本文探讨逐样本最小值计算问题，即所谓的在线最小值搜索。我们考虑了黄金分割搜索法的可行性，但通过量化分析证明二分法效率更高。在二分法中存在一个分层参数，用于调节每次子搜索的执行深度，其运作方式类似于深度优先搜索算法生成节点拓扑排序的过程。最后，我们提出利用调和测度的创新思路，这一方法在此前尚未被探索过。