Completely monotone intertwining relationsIn this talk, we introduce the concept of completely monotone intertwining relations as a strengthening of usual intertwining relations between Markov semi-groups. We proceed by providing some  interesting applications of this new idea which includes the characterization of ergodic constants and hypercontractivity estimates for non-self-adjoint semigroups.  We illustrate these results by presenting  several examples that have emerged from the recent literature: discrete-to-continuous interacting particle models, degenerate hypoelliptic Ornstein-Uhlenbeck processes, and  diffusion-to-jump  Jacobi processes.  This talk is based on a joint a work with L. Miclo (IMT Toulouse).