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：布朗运动或柯西过程的最小值分布可通过反射原理经典推导。本文探讨逐样本最小值搜索问题，即在线最小值搜索。我们考察黄金分割搜索法的可行性，并通过定量分析证明二分法具有更高效率。该二分法存在层级参数，用于调节各子搜索的深度，其机制类似于深度优先搜索生成节点拓扑排序的过程。最后，我们提出利用调和测度这一尚未被探索的新颖思路进行研究。