文摘
We investigate the process of eigenvalues of a fractional Wishart process defined by 1" class="mathmlsrc">1-s2.0-S0022123616303123&_mathId=si1.gif&_user=111111111&_pii=S0022123616303123&_rdoc=1&_issn=00221236&md5=208112aa82334ad7640fe5b09d3c845e" title="Click to view the MathML source">N=B⁎B, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter 1-s2.0-S0022123616303123&_mathId=si2.gif&_user=111111111&_pii=S0022123616303123&_rdoc=1&_issn=00221236&md5=e50e1958e49d040fee16783ec7126b60" title="Click to view the MathML source">H∈(1/2,1), we derive a stochastic differential equation in the 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 non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.
