Strong stationary times for one-dimensional diffusions

TL;DR

This paper establishes a necessary and sufficient condition for the existence of strong stationary times in ergodic one-dimensional diffusions, constructing them via dual processes and intertwinings, with applications to the Ornstein-Uhlenbeck process.

Contribution

It introduces a new characterization and construction method for strong stationary times in one-dimensional diffusions using dual processes and intertwinings.

Findings

Characterization of when strong stationary times exist for ergodic diffusions.

Construction of strong stationary times via dual processes and intertwinings.

Application to the Ornstein-Uhlenbeck process demonstrating the method's limitations.

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 $h$-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.