A path integral formulation of the Wright-Fisher process with genic selection

TL;DR

This paper introduces a novel path integral approach to analyze the Wright-Fisher process with selection, providing a perturbation series that accurately approximates transition densities, especially under weak selection conditions.

Contribution

The paper develops a path integral formalism for the Wright-Fisher process with selection, offering a new analytical tool beyond traditional PDE methods.

Findings

Perturbation series accurately approximates transition densities.

Feynman diagrams offer a probabilistic interpretation of selective events.

Method is arbitrarily accurate for any selection coefficient.

Abstract

The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient.