Strong Stationary Duality for Diffusion Processes

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...

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Bibliographic Details
Published in:Journal of theoretical probability 2016-12, Vol.29 (4), p.1298-1338
Main Authors: Fill, James Allen, Lyzinski, Vince
Format: Article
Language:English
Subjects:
Computer simulation
Markov analysis
Markov chains
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Spectral theory
Statistics
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Summary: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.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-015-0612-1