Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion

The non-Hermitian matrix-valued Brownian motion is the stochastic process of a random matrix whose entries are given by independent complex Brownian motions. The bi-orthogonality relation is imposed between the right and the left eigenvector processes, which allows for their scale transformations with an invariant eigenvalue process. The eigenvector-overlap process is a Hermitian matrix-valued process, each element of which is given by a product of an overlap of right eigenvectors and that of left eigenvectors. We derive a set of stochastic differential equations (SDEs) for the coupled system of the eigenvalue process and the eigenvector-overlap process and prove the scale-transformation invariance of the obtained SDE system. The Fuglede--Kadison (FK) determinant associated with the present matrix-valued process is regularized by introducing an auxiliary complex variable. This variable is necessary to give the stochastic partial differential equations (SPDEs) for the time-dependent random field defined by the regularized FK determinant and for its squared and logarithmic variations. Time-dependent point process of eigenvalues and its variation weighted by the diagonal elements of the eigenvector-overlap process are related to the derivatives of the logarithmic regularized FK-determinant random-field. We also discuss the PDEs obtained by averaging the SPDEs.