A pathwise interpretation of the Gorin-Shkolnikov identity

TL;DR

This paper provides a pathwise interpretation of the Gorin-Shkolnikov identity, linking the area under a normalized Brownian excursion and its local time through Jeulin's identity, revealing a Gaussian law connection.

Contribution

It introduces a novel pathwise perspective on the Gorin-Shkolnikov identity, emphasizing the role of Jeulin's identity in understanding the law of the involved stochastic quantities.

Findings

The area under a normalized Brownian excursion minus half its local time integral is Gaussian.

Jeulin's identity is key to the pathwise interpretation.

The law of the combined quantity is a centered Gaussian with variance 1/12.

Abstract

In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.