Intertwining diffusions and wave equations

We develop a general theory of intertwined diffusion processes of any dimension. Our main result gives an SDE characterization of all possible 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.