Trimming a Tree and the Two-Sided Skorohod Reflection

الأصل · EN

The h-trimming of a tree is a natural regularization procedure which consists in pruning the small branches of a tree: given h≥0, it is obtained by only keeping the vertices having at least one leaf above them at a distance greater or equal to h. The h-cut of a function f is the function of minimal total variation uniformly approximating the increments of f with accuracy h, and can be explicitly constructed via the two-sided Skorohod reflection of f on the interval [0,h]. In this work, we show that the contour path of the h-trimming of a rooted real tree is given by the h-cut of its original contour path. We provide two applications of this result. First, we recover a famous result of Neveu and Pitman, which states that the h-trimming of a tree coded by a Brownian excursion is distributed as a standard binary tree. In addition, we provide the joint distribution of this Brownian tree and its trimmed version in terms of the local time of the two-sided reflection of its contour path. As a second application, we relate the maximum of a sticky Brownian motion to the local time of its driving process.