A pathwise interpretation of the Gorin-Shkolnikov identity

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 $1\mathrm{/}12$. 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.