On intertwining relations between Ehrenfest, Yule and Ornstein-Uhlenbeck processes

Markovian intertwining relations between two Markov semigroups are related to the partial inclusion of the spectra of their generators, at least for finite ergodic processes. We check the limitations of this observation by investigating the Markov intertwining relations between the Ehrenfest, Yule and Ornstein-Uhlenbeck processes, whose spectra are all included into -${\mathbb{Z}}_{+}$. As a by-product, we offer a clarification of an intertwining relation found in Biane [2] between the Yule and the Ornstein-Uhlenbeck processes.