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

Joshua G. Schraiber

(Submitted on 29 Jul 2013)

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.

I’ve not had a chance to read it yet, but seeing the stuff about utilizing the deterministic path reminds me of Frank Norman’s work http://www.psych.upenn.edu/~norman/pubs.html. I think the paper http://www.psych.upenn.edu/~norman/Gaussian%20approximation.pdf takes a different diffusion limit for strong selection with small fluctuation. I know that’s not what you are doing [I think] but it might be worth a look.