publication . Preprint . Article . 2014

Strong Stationary Duality for Diffusion Processes

James Allen Fill; Vince Lyzinski;
Open Access English
  • Published: 23 Feb 2014
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...
Subjects
free text keywords: Mathematics - Probability, Statistics, Probability and Uncertainty, Statistics and Probability, General Mathematics, Statistical physics, Mathematical analysis, Mathematics, Markov chain, Cutoff, Calculus, Spectral theory, Duality (optimization), Limiting
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[S(x) − S(η)] M (dη), Σ(l) := [M (x) − M (η)] dS(η) π(y) Π(x) −1 M (y) M (y) s(y) dy. s(y)M (y) dy − s(y)M (y) dy < ∞ − M (y) dy s(y)[M (y) − M (x)] dy = Σ(1) = ∞ s(y) M (y) − M (1) s(y)[M (1) − M (y)] dy = N (1) < ∞. A − α A − Proof. Let f ∈ F [0, 1]. By Remark 3.12, we have for all t > 0 that (Πt, f ) = (Πt∗, Λf ). Therefore, writing Rt = dΠt/dΠ as usual, we have Z Z 1 + c2λn,1tn

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