Representation Theorems for Interacting Moran Models, Interacting Fisher-Wrighter Diffusions and Applications
Representation Theorems for Interacting Moran Models, Interacting Fisher-Wrighter Diffusions and Applications
复制标题
相互作用 Moran 模型、相互作用 Fisher-Wrighter 扩散和应用的表示定理
DOI：
10.1214/ejp.v10-290
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发表时间：
2005
影响因子：
1.4
通讯作者：
A. Winter
中科院分区：
文献类型：
--
作者：
A. Greven;V. Limic;A. Winter
我们考虑空间相互作用的Moran模型及其扩散极限，即相互作用的Fisher-Wright扩散。Moran模型是一个空间种群模型，不同类型的个体位于由阿贝尔群的元素给定的地点。该系统的动力学由个体在地点之间的独立迁移和每个地点的重新安置机制组成，即，成对的个体被新的个体所取代，其中每个新的个体采取从父代个体对中随机选择的个体的类型。相互作用的Fisher-Wright扩散收集在每个站点无限多个个体的限制下针对单独站点评估的类型子集的相对频率。一个是感兴趣的类型配置以及时间-空间演变的系谱，编码在所谓的历史过程。本文的第一个目标是分析表征的历史过程中，这两个模型的解决方案，适定的鞅问题和相应的对偶理论的发展。为此，我们通过所谓的向下看过程将历史上的Fisher-Wright扩散和历史上的Moran模型联系起来。也就是说，对于任何固定的时间，历史莫兰模型与增加的粒子强度和粒子表示的限制历史相互作用的费舍尔-赖特扩散的集合提供在一个和相同的概率空间。这导致了空间相互作用的莫兰模型之间的强形式的对偶性，一方面是相互作用的Fisher-Wright扩散，另一方面是合并的随机游动，这扩展了经典的弱形式的矩对偶性。我们的第二个目标是表明，这种表示可以用来获得新的结果的长期行为，特别是（i）的结构的平衡，和平衡的历史过程，和（ii）的行为，我们的模型在大，但有限的网站空间相比，我们的模型在无限网站空间。在这里，所谓的有限系统计划建立空间相互作用的莫兰模型，这意味着通过向下看表示也是已知的结果，相互作用的费舍尔-赖特扩散。此外，适当的版本的有限系统计划的水平上的历史过程的新开发和验证。从长远来看，所提供的向下看表示的目的是回答有关相互作用的费舍尔-赖特扩散的更精细的路径属性的问题。
We consider spatially interacting Moran models and their diffusion limit which are interacting Fisher-Wright diffusions. The Moran model is a spatial population model with individuals of different type located on sites given by elements of an Abelian group. The dynamics of the system consists of independent migration of individuals between the sites and a resampling mechanism at each site, i.e., pairs of individuals are replaced by new pairs where each newcomer takes the type of a randomly chosen individual from the parent pair. Interacting Fisher-Wright diffusions collect the relative frequency of a subset of types evaluated for the separate sites in the limit of infinitely many individuals per site. One is interested in the type configuration as well as the time-space evolution of genealogies, encoded in the so-called historical process. The first goal of the paper is the analytical characterization of the historical processes for both models as solutions of well-posed martingale problems and the development of a corresponding duality theory. For that purpose, we link both the historical Fisher-Wright diffusions and the historical Moran models by the so-called look-down process. That is, for any fixed time, a collection of historical Moran models with increasing particle intensity and a particle representation for the limiting historical interacting Fisher-Wright diffusions are provided on one and the same probability space. This leads to a strong form of duality between spatially interacting Moran models, interacting Fisher-Wright diffusions on the one hand and coalescing random walks on the other hand, which extends the classical weak form of moment duality for interacting Fisher-Wright diffusions. Our second goal is to show that this representation can be used to obtain new results on the long-time behavior, in particular (i) on the structure of the equilibria, and of the equilibrium historical processes, and (ii) on the behavior of our models on large but finite site space in comparison with our models on infinite site space. Here the so-called finite system scheme is established for spatially interacting Moran models which implies via the look-down representation also the already known results for interacting Fisher-Wright diffusions. Furthermore suitable versions of the finite system scheme on the level of historical processes are newly developed and verified. In the long run the provided look-down representation is intended to answer questions about finer path properties of interacting Fisher-Wright diffusions.