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

## TL;DR

This paper explores the spectral relationships and intertwining relations among Ehrenfest, Yule, and Ornstein-Uhlenbeck processes, clarifying existing connections and examining spectral inclusion limitations in Markovian intertwining.

## Contribution

It investigates the spectral inclusion and intertwining relations among these processes, providing new insights and clarifications on their spectral and probabilistic connections.

## Key findings

- Spectra of the processes are all included in -Z_+.
- Limitations of spectral inclusion in Markov intertwining are analyzed.
- Clarification of an existing intertwining relation between Yule and Ornstein-Uhlenbeck processes.

## Abstract

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.