Strong stationary times for one-dimensional diffusions

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 $h$-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 in the total variation sense.

Mots clés: Strong (stationary) time; Bessel process; Ornstein-Uhlenbeck process; spectral decomposition and quasi-stationary measure.; Ergodic one-dimensional diffusion; Intertwining; Dual process; Evolving segments; Explosion time; Spectral decomposition and quasi-stationary measure

Domaines: Probabilités [math.PR]

Laurent Miclo. Strong stationary times for one-dimensional diffusions. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017, ⟨10.1214/16-aihp745⟩. ⟨hal-00909133v2⟩