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 paper, 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.
"A pathwise interpretation of the Gorin-Shkolnikov identity." Electron. Commun. Probab. 21 1 - 6, 2016. https://doi.org/10.1214/16-ECP10