立方复形上随机游走到布朗运动的收敛
种群与进化
2019-05-23 v4 概率论
摘要
立方复形是通过将单位立方体粘合在一起构成的度量空间，其构造方式类似于单纯复形的构造。我们在此类空间上构造布朗运动，定义随机游走，并证明随机游走的转移核收敛于布朗运动的转移核。证明涉及将标准立方体上布朗样本路径的分布拉回至该复形，并将其与复形中立方体间游走的分布相结合。主要应用在于分析进化树集合：若干树空间是立方复形，我们简要描述了在此背景下的结果及一些应用。我们的结果容易推广到一类多面体复形，其中每个最大维度的胞腔与给定固定多面体等距。
引用
@article{arxiv.1508.02906,
title = {Convergence of random walks to Brownian motion on cubical complexes},
author = {Tom M. W. Nye},
journal= {arXiv preprint arXiv:1508.02906},
year = {2019}
}
备注
14 pages, 2 figures. The results in the original submission have been changed substantially. In particular, the main theorem has been generalized to apply to a wide class of cubical complexes rather than Billera-Holmes-Vogtmann tree space alone. This simplifies some parts of the proof, although the main ideas are the same. Tree space is now dealt with as a special example in Section 5