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
26 references, page 1 of 2

[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

[7] E. B. Dynkin. Markov processes. Vols. I, II, volume 122 of Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, B¨ande 121. Academic Press Inc., Publishers, New York, 1965.

[8] S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. Characterization and convergence.

[9] J. A. Fill. Strong stationary duality for continuous-time Markov chains. Part I: Theory. Journal of Theoretical Probability, 5:45-70, 1992. 10.1007/BF01046778.

[10] J. A. Fill. An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab., 8(1):131-162, 1998. [OpenAIRE]

[11] J. A. Fill. The passage time distribution for a birth-and-death chain: strong stationary duality gives a first stochastic proof. J. Theoret. Probab., 22(3):543-557, 2009. [OpenAIRE]

[12] J. A. Fill and J. Kahn. Comparison inequalities and fastest-mixing markov chains. Annals of Applied Probability, 23(5):1778-1816, 2013. [OpenAIRE]

[13] J. A. Fill, M. Machida, D. J. Murdoch, and J. S. Rosenthal. Extension of Fill's perfect rejection sampling algorithm to general chains. Random Structures Algorithms, 17(3-4):290-316, 2000. Special issue: Proceedings of the Ninth International Conference “Random Structures and Algorithms” (Poznan, 1999).

[14] JamesAllen Fill and Vince Lyzinski. Hitting times and interlacing eigenvalues: A stochastic approach using intertwinings. Journal of Theoretical Probability, pages 1-28, 2012. [OpenAIRE]

[15] R. Holley and D. Stroock. Dual processes and their application to infinite interacting systems. Adv. in Math., 32(2):149-174, 1979. [OpenAIRE]

[16] K. Itoˆ and H. P. McKean, Jr. Diffusion processes and their sample paths. SpringerVerlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125.

[17] S. Karlin and J. McGregor. Coincidence properties of birth and death properties. Pacific J. Math., 9:1109-1140, 1959. [OpenAIRE]

[18] S. Karlin and H. M. Taylor. A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981.

[19] J. T. Kent. Eigenvalue expansions for diffusion hitting times. Z. Wahrsch. Verw. Gebiete, 52(3):309-319, 1980.

[20] F. B. Knight. Essentials of Brownian motion and diffusion, volume 18 of Mathematical Surveys. American Mathematical Society, Providence, R.I., 1981.

26 references, page 1 of 2
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue