  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.
