WRIGHT-FISHER DIFFUSION WITH NEGATIVE MUTATION RATES
WRIGHT-FISHER DIFFUSION WITH NEGATIVE MUTATION RATES
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DOI：
10.1214/11-aop704
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发表时间：
2013-03-01
影响因子：
2.3
通讯作者：
Pal, Soumik
中科院分区：
文献类型：
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作者：
Pal, Soumik
我们研究了一个家庭的n维扩散，在单位单形的向量与非负坐标，加起来一个值。这些过程满足的随机微分方程类似于经典的赖特-费舍尔扩散，除了“突变率”现在是非正的。这个模型，建议奥尔德斯，出现在研究的一个约束扩散极限的马尔可夫链上的分支图。这些模型的显著特征是边界没有反射，一旦它到达边界，我们就杀死这个过程。我们推导出明确的退出分布的单纯形和概率界的退出时间。我们还证明，这些过程可以被看作是一个“随机时间反转”的Wright-Fisher过程的尺寸增加，并在一个随机时间的条件。在我们的证明中的一个关键思想是使用某些一维扩散称为贝塞尔平方过程的负维，这是最近推出的Going-Jaeschke和Yor的斜积建设。
We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher diffusions, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Going-Jaeschke and Yor.