Multilevel Dyson Brownian motions via the superposition principle

Abstract

Multilevel Dyson Brownian motions (MDBMs) combine Dyson Brownian motions of different dimensions into a single process in a canonical way. This paper completes the theory of MDBMs for $\beta\ge2$. Specifically, we use the superposition principle of Figalli and Trevisan to construct the MDBMs for all $\beta>2$ in a unified manner. This also extends their stochastic differential equation representation, first discovered by Gorin and Shkolnikov, to all $\beta>2$ and proves the uniqueness of the MDBMs for all $\beta>2$. Finally, we show that their limit as $\beta\downarrow2$ is given by the $\beta=2$ MDBM, commonly referred to as the Warren process.