Hermitian Functional Representation of Free L\'evy Processes

A functional representation of free L'evy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued L'evy processes. This is accomplished by proving functional asymptotics of their empirical spectral processes towards the law of a free L'evy processes. This result recovers a functional version of Wigner’s theorem and introduces a functional version of Marchenko-Pastur’s theorem providing the free Poisson process as the noncommutative limit process.