The Gorin-Shkolnikov Identity and Its Random Tree Generalization

In a recent pair of papers, Gorin and Shkolnikov (Ann Probab 46: 2287-2344, 2018) and Hariya (Electron Commun Probab 21: 6, 2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance 1/12. Lamarre and Shkolnikov generalized this to Brownian bridges (Lamarre and Shkolnikov inAnn Inst Henri Poincare Probab Stat 55: 1402-1438, 2019) and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on n vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.

Automated summary

The papers demonstrated that the area under a normalized Brownian motion minus one-half the integral of its total local time squared is a centered normal random variable with variance 1/12. This result was extended to Brownian bridges with combinatorial interpretation using random forests and process level generalization for a certain infinite forest model. Analogous results were also shown for various related models using stochastic calculus.