The Dyson Brownian minor process

Abstract

Consider an n × n Hermitean matrix valued stochastic process {Ht}t≥0 where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the k×k minors in the upper left corner of Ht. Projecting this process to a space-like path leads to a determinantal process for which we compute the kernel. This kernel contains the well known GUE minor kernel, and the Dyson Brownian motion kernel as special cases. In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier's bead kernel. We also compute the kernel for a process of intertwined Brownian motions introduced by Warren. That too is a determinantal process along space-like paths.