Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Strong stationary times for one-dimensional diffusions

Laurent Miclo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A necessary and sufficient condition is obtained for the existence of strong stationary times for ergodic one-dimensional diffusions, whatever the initial distribution. The strong stationary times are constructed through intertwinings with dual processes, in the Diaconis–Fill sense, taking values in the set of segments of the extended line $\mathbb{R}\sqcup\{-\infty,+\infty\}$. They can be seen as natural Doob transforms of the extensions to the diffusion framework of the evolving sets of Morris–Peres. Starting from a singleton set, the dual process begins by evolving into true segments in the same way a Bessel process of dimension 3 escapes from 0. The strong stationary time corresponds to the first time the full segment $[-\infty,+\infty]$ is reached. The benchmark Ornstein–Uhlenbeck process cannot be treated in this way; it will nevertheless be seen how to use other strong times to recover its optimal exponential rate of convergence to equilibrium in the total variation sense.

Résumé

Une condition nécessaire et suffisante est obtenue pour l’existence de temps fort de stationnarité, quelque soit la condition initiale. Les temps forts de stationnarité sont construits par le biais d’entrelacements avec des processus duaux, au sens de Diaconis–Fill, prenant leurs valeurs dans l’ensemble des segments de la droite étendue $\mathbb{R}\sqcup\{-\infty,+\infty\}$. Ils peuvent être vus comme des transformées de Doob d’extensions au cadre diffusif des ensembles évoluants de Morris–Peres. Partant d’un singleton, le processus dual commence par évoluer en segments de la même manière qu’un processus de Bessel de dimension 3 s’échappe de 0. Le temps fort de stationnarité correspond au premier temps d’atteinte de $[-\infty,+\infty]$. Le processus d’Ornstein–Uhlenbeck ne peut pas être traiter de la sorte, il est toutefois possible d’utiliser d’autres temps forts pour retrouver son taux exponentiel optimal de convergence à l’équilibre en variation totale.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 957-996.

Dates
Received: 12 June 2015
Revised: 16 January 2016
Accepted: 7 February 2016
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897753

Digital Object Identifier
doi:10.1214/16-AIHP745

Mathematical Reviews number (MathSciNet)
MR3634282

Zentralblatt MATH identifier
1367.60101

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 47A10: Spectrum, resolvent 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60E15: Inequalities; stochastic orderings

Keywords
Strong (stationary) time Ergodic one-dimensional diffusion Intertwining Dual process Evolving segments Explosion time Bessel process Ornstein–Uhlenbeck process Spectral decomposition and quasi-stationary measure

Citation

Miclo, Laurent. Strong stationary times for one-dimensional diffusions. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 957--996. doi:10.1214/16-AIHP745. https://projecteuclid.org/euclid.aihp/1491897753


Export citation

References

  • [1] D. Aldous and P. Diaconis. Strong uniform times and finite random walks. Adv. in Appl. Math. 8 (1) (1987) 69–97.
  • [2] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Société Mathématique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.
  • [3] D. Bakry. L’hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability Theory (Saint-Flour, 1992) 1–114. Lecture Notes in Math. 1581. Springer, Berlin, 1994.
  • [4] D. Bakry, I. Gentil and M. Ledoux. Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham, 2014.
  • [5] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
  • [6] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1) (1999) 1–28.
  • [7] P. Carmona, F. Petit and M. Yor. Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoam. 14 (2) (1998) 311–367.
  • [8] L. Cheng and Y. Mao. Passage time distribution for one-dimensional diffusion processes. Preprint, 2013.
  • [9] L. Cheng and Y. Mao. Eigentime identity for one-dimensional diffusion processes. J. Appl. Probab. 52 (1) (2015) 224–237.
  • [10] P. Collet, S. Martínez and J. San Martín. Markov Chains, Diffusions and Dynamical Systems Quasi-Stationary Distributions. Probability and Its Applications (New York). Springer, Heidelberg, 2013.
  • [11] C. Dellacherie and P. Meyer. Probabilités et Potentiel, revised edition. Chapitres V à VIII Théorie des Martingales [Martingale Theory]. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics] 1385. Hermann, Paris, 1980.
  • [12] D. Persi and J. A. Fill. Strong stationary times via a new form of duality. Ann. Probab. 18 (4) (1990) 1483–1522.
  • [13] D. Persi and L. Miclo. On times to quasi-stationarity for birth and death processes. J. Theoret. Probab. 22 (3) (2009) 558–586.
  • [14] D. Persi and L. Saloff-Coste. Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16 (4) (2006) 2098–2122.
  • [15] L. E. Dubins. On a theorem of Skorohod. Ann. Math. Stat. 39 (1968) 2094–2097.
  • [16] J. A. Fill. An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8 (1) (1998) 131–162.
  • [17] J. A. Fill and J. Kahn. Comparison inequalities and fastest-mixing Markov chains. Ann. Appl. Probab. 23 (5) (2013) 1778–1816.
  • [18] J. A. Fill and V. Lyzinski. Strong stationary duality for diffusion processes. J. Theoret. Probab. 29 (4) (2016) 1298–1338.
  • [19] T. E. Huillet and S. Martinez. On Möbius duality and coarse-graining. J. Theoret. Probab. 29 (1) (2016) 143–179.
  • [20] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland Mathematical Library 24. North-Holland Publishing Co., Amsterdam, 1989.
  • [21] S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981.
  • [22] J. Kent. Time-reversible diffusions. Adv. in Appl. Probab. 10 (4) (1978) 819–835.
  • [23] M. Ledoux. Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994) 165–294. Lecture Notes in Math. 1648. Springer, Berlin, 1996.
  • [24] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics. Springer, Berlin, 2005. Reprint of the 1985 original.
  • [25] P. Lorek and R. Szekli. Strong stationary duality for Möbius monotone Markov chains. Queueing Syst. 71 (1–2) (2012) 79–95.
  • [26] Y. Mao. Eigentime identity for transient Markov chains. J. Math. Anal. Appl. 315 (2) (2006) 415–424.
  • [27] L. Miclo. On ergodic diffusions on continuous graphs whose centered resolvent admits a trace. J. Math. Anal. Appl. 437 (2) (2016) 737–753.
  • [28] M. Ben and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2) (2005) 245–266.
  • [29] S. Pal and M. Shkolnikov. Intertwining diffusions and wave equations. Preprint, 2013. Available at http://arxiv.org/abs/1306.0857.
  • [30] J. W. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 (3) (1975) 511–526.
  • [31] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [32] C. Rogers and J. W. Pitman. Markov functions. Ann. Probab. 9 (4) (1981) 573–582.