Markov intertwining relations: from card shuffles to stochastic variants of mean curvature flows

Abstract

Consider a deck of N cards that is shuffled in the following way: take the card at the top and put it at a uniformly random position in the deck. For how long this procedure must be repeated in order to mix the deck? Aldous et Diaconis (1986) gave a simple answer to this question by resorting to a strong stationary time and to an intertwining dual. After explaining these notions and why they matter, we will see how they are also underlying a theorem of Pitman (1975) on the links between the Brownian motion and the Bessel-3 process (which is the norm of a Brownian motion in dimension 3). This kind of constructions will be generalized to the stochastic evolution of forms in Riemannian manifolds, to visualize how the randomness of elliptic diffusions is propagating in the state space.