Consecutive minors for Dyson's Brownian motions

In 1962, Dyson (1962) introduced dynamics in random matrix models, in particular into GUE (also for β=1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: β=1,2,4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for β=2,4.