  We develop the theory of strong stationary duality for diffusion processes on
compact intervals. We analytically derive the generator and boundary behavior
of the dual process and recover a central tenet of the classical Markov chain
theory in the diffusion setting by linking the separation distance in the
primal diffusion to the absorption time in the dual diffusion. We also exhibit
our strong stationary dual as the natural limiting process of the strong
stationary dual sequence of a well chosen sequence of approximating
birth-and-death Markov chains, allowing for simultaneous numerical simulations
of our primal and dual diffusion processes. Lastly, we show how our new
definition of diffusion duality allows the spectral theory of cutoff phenomena
to extend naturally from birth-and-death Markov chains to the present diffusion
context.
