Wednesday, 11 December
Title: On hyperboundedness and spectrum of Markov operators.
Abstract: Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $mu$ on a general measurable space. We will show that if $M$ is bounded from ${cal L}^2(mu)$ to ${cal L}^p(mu)$, where $p>2$, then it admits a spectral gap.This result answers positively a conjecture raised by H{o}egh-Krohn and Simon in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan. In general there is no quantitative link between hyperboundedness and spectral gap (except in the situation already investigated by Wang), but there is one with another eigenvalue.In addition, the usual Cheeger inequalities will be extended to higher eigenvalues in the compact Riemannian setting.
Thursday, 12 December
Title: Strong stationary times for one-dimensional diffusions.
Abstract: A strong stationary time associated to an ergodic Markov process $X$ is a stopping time $tau$ which is independent from the stopped position $X_tau$ and such that $X_tau$ is distributed according to the underlying invariant measure. We will present a necessary and sufficient condition ifor 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 setof 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.