## Groupe de travail Bio-Maths

**Free energy functional method and its applications to the Wright-Fisher models**

Dat Tran

Thursday 07 June 2018 15:30 - -

**Résumé :**

In this talk, I will first give out a very brief introduction to Wright-Fisher models, the most popular models in mathematical population genetics, which can be approximated as a diffusion process in a simplex. The conditional density function of the process is known as the solution of a forward Kolmogorov equation (also known by physicists as a Fokker-Planck equation). The difficulty is that the generator is singular and the boundary of the domain is not smooth. I will then discuss about a concept from stochastic mechanism so-called free energy functional and how to apply it in this setting. In particular, I shall provide a necessary and sufficient condition in terms of mutation and selection coefficients under which the Wright-Fisher diffusion process will possess a free energy functional. This then also is a necessary and sufficient condition for the existence of a unique reversible probability density with respect to the Lebesgue measure. Moreover, by using Bakry-Emery techniques, I will show that under an additional condition, the flow of probability densities exponentially converges to the reversible one in some distance such as total variation. Some open problems will also be discussed. (This is the joint work with Juergen Jost and Julian Hofrichter.)

**Liens :**