Mathematical Sciences: Some Aspects of Markov Processes in Population Genetics

本研究将研究数理种群遗传学中的随机模型。第一组问题涉及测量值扩散过程，它推广了群体遗传学中常用的k等位基因扩散模型。研究者将研究选择性无限多等位基因模型的遍历性，双位点模型的性质，以及具有重组的无限多位点模型的存在性。第二组问题涉及具有两个时间尺度的遗传模型的高斯扩散近似，以及与聚结相关的祖先马尔可夫链的收敛性。

This research will study stochastic models in mathematical population genetics. The first set of problems concerns measure- valued diffusion processes that generalize the usual K-allele diffusion model of population genetics. The investigator will study the ergodicity of the selective infinitely-many-alleles model, properties of the two-locus model, and existence of the infinitely-many-sites model with recombination. The second set of problems relates to Gaussian diffusion approximations of genetic models with two time scales, and convergence of an ancestral Markov chain related to the coalescent.