Benjamin Budway | Searcharxiv

arXiv2024

Multilevel Dyson Brownian motions via the superposition principle

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.

math.PR↗

arXiv2013

Intertwining diffusions and wave equations

We develop a general theory of intertwined diffusion processes of any dimension. Our main result gives an SDE construction of intertwinings of diffusion processes and shows that they correspond to nonnegative solutions of hyperbolic partial differential equations. For example, solutions of the classical wave equation correspond to the intertwinings of two Brownian motions. The theory allows us to unify many older examples of intertwinings, such as the process extension of the beta-gamma algebra, with more recent examples such as the ones arising in the study of two-dimensional growth models. We also find many new classes of intertwinings and develop systematic procedures for building more complex intertwinings by combining simpler ones. In particular, `orthogonal waves' combine unidimensional intertwinings to produce multidimensional ones. Connections with duality, time reversals and Doob's h-transforms are also explored.

math.PR↗