  We define Laguerre and Jacobi analogues of the Warren process. That is, we
construct local dynamics on a triangular array of particles so that the
projections to each level recover the Laguerre and Jacobi eigenvalue processes
of K\"onig-O'Connell and Doumerc and the fixed time distributions recover the
joint distribution of eigenvalues in multilevel Laguerre and Jacobi random
matrix ensembles. Our techniques extend and generalize the framework of
intertwining diffusions developed by Pal-Shkolnikov. One consequence is the
construction of particle systems with local interactions whose fixed time
distribution recovers the hard edge of random matrix theory. An appendix by
Andrey Sarantsev establishes strong existence and uniqueness for solutions to
SDER's satisfied by these processes.
