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.


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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