On the non-commutative fractional Wishart process

详细信息    查看全文

• 作者：Juan Carlos Pardo^a ; ^{jcpardo@cimat.mx" class="auth_mail" title="E-mail the corresponding author} ; José ; -Luis Pé ; rez^a ; ^b ; ^{jluis.garmendia@cimat.mx" class="auth_mail" title="E-mail the corresponding author} ; Victor Pé ; rez-Abreu^a ; ^{pabreu@cimat.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional Wishart matrix process ; Measure valued process ; Young integral ; Fractional calculus
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：272
• 期：1
• 页码：339-362
• 全文大小：434 K

文摘

We investigate the process of eigenvalues of a fractional Wishart process defined by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=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^⁎Bclass="mathContainer hidden">class="mathCode">$N=B^{⁎}B$, 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=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)class="mathContainer hidden">class="mathCode">$H\in (1/2,1)$, 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.