Fleming–Viot Processes in Population Genetics

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include: 1. Introduction. 2. Some measure-valued Markov chains. 2.1. A diploid model. 2.2. The Wright–Fisher model. 2.3. A Moran model. 3. The Fleming–Viot process: characterization. 4. Convergence. 5. Ergodicity. 6. An infinite particle system. 7. Bounded mutation operators. 8. Reversibility. 9. Examples. 9.1. Continuous-state stepwise-mutation model. 9.2. Infinitely-many-neutral-alleles model. 9.3. Infinitely-many-neutral-alleles model with ages. 9.4. Two-locus model with recombination. 9.5. n-locus model with gene conversion. 9.6. Infinitely-many-sites model without recombination. 9.7. Infinitely-many-neutral-alleles model with allelic genealogies.