Functionals of the Free Brownian Bridge

TL;DR

This paper explores the distributions of key functionals of the free Brownian bridge, revealing their free infinite divisibility and providing series representations analogous to classical Fourier series.

Contribution

It introduces new series representations of the free Brownian bridge using free semicircular variables, extending classical Fourier methods to free probability.

Findings

The -norm of the free Brownian bridge is freely infinitely divisible.

The second component of its signature is characterized probabilistically.

The Le9vy area functional is also shown to be freely infinitely divisible.

Abstract

We discuss the distributions of three functionals of the free Brownian bridge: its $\L^2$-norm, the second component of its signature and its L\'evy area. All of these are freely infinitely divisible. We introduce two representations of the free Brownian bridge as series of free semicircular random variables are used, analogous to the Fourier representations of the classical Brownian bridge due to \ts{L\'evy} and \ts{Kac}.