# Intertwining of the Wright-Fisher diffusion

## TL;DR

This paper uses intertwining Markov process techniques to couple the Wright-Fisher diffusion with a pure birth process, revealing insights into absorption times and diffusion behavior.

## Contribution

It introduces a novel coupling of the Wright-Fisher diffusion with a pure birth process using intertwining, linking absorption and explosion times.

## Key findings

- Absorption time of the diffusion equals explosion time of the birth process.
- Diffusion initially reluctant to be absorbed, then increasingly compelled.
- Coupling provides new probabilistic interpretation of diffusion behavior.

## Abstract

It is known that the time until a birth and death process reaches a certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. Their coupling is of a special type called intertwining of Markov processes. We apply this technique to couple the Wright-Fisher diffusion with reflection at 1/2 and a pure birth process. We show that in our coupling the time of absorption of the diffusion is a.s. equal to the time of explosion of the pure birth process. The coupling also allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed.