Free Jacobi Process

Abstract

Using a matrix approach, we define the free Jacobi process as the limit of the complex Jacobi matrix process. The we derive a free SDE which is analogous to its classical counterpart. To proceed, we prove that fro suitable parameters the process remains injective if it is initially injective and then use the polar decomposition. In the stationnary case, this will be easily deduced from the explicit expression of the spectral measure. In the general setting we derive a recurrence formula for the moments. Moreover, a p. d. e. for the Cauchy transform of the law is given.