# The transition distribution of a sample from a Wright-Fisher diffusion with general small mutation rates

## TL;DR

This paper derives an approximation for the transition distribution of a sample from a Wright-Fisher diffusion with small mutation rates using a coalescent approach, addressing a previously unknown transition density.

## Contribution

It provides the first known approximation of the transition density for Wright-Fisher diffusion with general small mutation rates.

## Key findings

- Derived a coalescent-based approximation for the transition distribution.
- Established an explicit form for the transition density in the small mutation regime.
- Enhanced understanding of genetic variation dynamics in population genetics.

## Abstract

The transition distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree of the sample up to the most recent common ancestor with additional mutations occurring on the lineage from the most recent common ancestor to the time origin if complete coalescence occurs before the origin. The sampling distribution leads to an approximation for the transition density in the diffusion with small mutation rates. This new solution has interest because the transition density in a Wright-Fisher diffusion with general mutation rates is not known.