Strong Stationary Duality for Diffusion Processes

TL;DR

This paper develops a theory of strong stationary duality for diffusion processes, linking separation distance to absorption time, and extends spectral cutoff phenomena from Markov chains to diffusions.

Contribution

It introduces a new framework for strong stationary duality in diffusion processes, including generator derivation, boundary behavior, and numerical simulation methods.

Findings

Dual process generator and boundary behavior derived.

Separation distance linked to absorption time in dual.

Spectral cutoff phenomena extended to diffusions.

Abstract

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.