## Strong stationary times for one-dimensional diffusions.(English. French summary)Zbl 1367.60101

Summary: 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.

### MSC:

 60J60 Diffusion processes 60J35 Transition functions, generators and resolvents 60E15 Inequalities; stochastic orderings 37A30 Ergodic theorems, spectral theory, Markov operators 47A10 Spectrum, resolvent
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