Brownian motion on the complex general group and the non-compact Jacobi process

Abstract

In this paper, we introduce and study a new non Hermitian matrix-valued process built from truncations of Brownian motion on complex positive-definite matrices and of its inverse. We prove that its eigenvalue process is valued in $(1,+\infty )$ and that it forms a non-compact Jacobi particle system. Surprisingly, the same particle system occurs in \cite{BDW} in connection with Brownian motion on a non-compact complex Grassmann manifold. In this respect, we supply evidences showing that this occurrence holds true only at the spectral level and not on the corresponding matrix diffusions. We then study the large-size limit of our matrix model and prove results parallel to those obtained by the second author for the free Jacobi process.

Revised version