Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

A Markov intertwining relation between two Markov processes X and Y is a weak similitude relation G\Lambda = \Lambda L between their generators L and G, where \Lambda is a transition kernel between the underlying state spaces. This notion is an important tool to deduce quantitative estimates on the speed of convergence to equilibrium of X via strong stationary times when Y is absorbed, as shown by the theory of Diaconis and Fill for finite state spaces. In this talk we will only consider processes Y taking as values some subsets of the state space of X. Our goal is to present extensions of the above method to elliptic diffusion processes on differentiable manifolds, via stochastic modifications of mean curvature flows.  We will see that Pitman's theorem about the intertwining relation between the Brownian motion and the Bessel-3 process is curiously ubiquitous in this approach.  It  even serves  as an inspiring guide to construct couplings associated to finite Markov intertwining relations via random mappings, in the spirit of the coupling-from-the-past algorithm of Propp and Wilson and of the evolving sets of Morris and Peres.