Reading Group Topic: Diffusion theory in genetics

Time: Term 2, Wednesdays 12:00–13:00 in room MB 0.08 and online. The first meeting will be in week 2 (19th Jan 2022).

About: Diffusions are continuous Markov processes in which both time and space are treated continuously. They arise as models in all sorts of scientific disciplines: in population genetics they serve as a fundamental model of evolution, describing changing allele frequencies over time in a large population. This reading group will review the main theory of the so-called Wright–Fisher diffusion model of evolution, with a particular focus on its boundary behaviour. We will also focus on excursion theory for diffusions—the specialisation of this theory to the Wright-Fisher diffusion seems not to be well-developed and could be the focus of a PhD project. Although we will focus on theoretical aspects, there are also many interesting statistical questions that could lead to a PhD project, so we expect this group might also to be of interest to students interested in statistical inference from stochastic processes wishing to develop their background knowledge of stochastic processes.

We will start by covering the introductory material in Etheridge (2011) “Some Mathematical models in population genetics”, Chapters 3–5; see also Karlin & Taylor (1981) “A second course in stochastic processes”, Chapter 15.

Depending on progress and interest, we might move on to some more specialised topics including those covered in the following:

• Rogers (1989) “A guided tour through excursions”, Bull. London Math. Soc. 21:305–341.
• Bertoin (1996) “Levy Processes”, Chapter 4.
• Pitman & Yor (2003) “Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches”, Bernoulli 9:1–24.
• Pitman & Yor (2007) “Itô’s excursion theory and its applications”, Japan J. Math 2:83–96.
• Pardoux (2009) “Probabilistic models of population genetics”, Lecture notes.

Organisers: Paul Jenkins, Jere Koskela, Dario Spano, Jaromir Sant.

Please contact one of the organisers if you are interested in taking part.