On the Free Fractional Wishart Process

TL;DR

This paper studies the eigenvalue dynamics of a fractional Wishart process driven by fractional Brownian motion, proving non-collision, deriving SDEs, and establishing a limit theorem for the empirical eigenvalue distribution.

Contribution

It introduces a stochastic calculus framework for fractional Wishart eigenvalues, proves non-collision, derives SDEs, and characterizes the limit as a free fractional Wishart process.

Findings

Eigenvalues do not collide with probability one.

Derived SDEs for eigenvalues with Hurst parameter H in (1/2, 1).

Established a limit theorem for the empirical eigenvalue distribution.

Abstract

We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and P\'erez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the free fractional Wishart process which constitutes the family of fractional dilations of the free Poisson distribution.