Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Strong stationary times for one-dimensional diffusions

Laurent Miclo

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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