State Dependent Multitype Spatial Branching Processes and their Longtime Behavior
State Dependent Multitype Spatial Branching Processes and their Longtime Behavior
复制标题
状态相关的多类型空间分支过程及其长期行为
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
A. Greven
中科院分区：
文献类型：
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作者：
D. Dawson;A. Greven
本文主要研究空间分量(群体)由一个可数群索引的空间多类型分枝系统，例如$Z^d$或层次群。作为类型空间，我们允许连续体，并将一个群体中的种群描述为类型空间上的度量。系统的空间组件通过迁移进行交互。与经典的关于分枝种群不同家族演化的独立性假设不同，我们通过个体的状态依赖分枝率和个体的状态依赖的平均后代来引入家庭之间的相互作用。然而，对于大多数结果，我们认为这项工作是关键情况。所考虑的系统是由单个粒子的分枝率和子代平均值引起的单个粒子在可数群上的临界多类型分枝随机游动的扩散极限，它依赖于所讨论粒子所在位置的总种群。本文的主要目的是构造所讨论的度量值扩散，用良定的鞅问题刻画它们，并最终确定它们的长期性态，其中包含一些新的特征。进一步，我们确定了系统的两个泛函的动力学，即各点的总质量过程和群体中不同类型的相对权重分别作为相互作用的扩散系统的时间非齐次Fleming-Viot过程。这需要对总质量过程的路径属性进行详细的分析。除了上述相互作用的度量值过程系统外，我们还通过适定的鞅问题构造了相应的历史过程。历史进程包括关于家庭结构的信息，即个人之间不同程度的关系。在临界情况下，证明了过程和历史过程的遍历定理，以及相应的总质量泛函和相对权泛函。在对称化运动是经常性的瞬时运动的情况下，长时间行为有质的不同。我们在一种情况下看到了局部灭绝，在另一种情况下看到了诚实的均衡。这整个计划需要开发一些新的技术，这些技术应该在更广泛的背景下引起兴趣。对于具有多维分量的系统，这类工具是在具有奇异性和耦合的随机波动介质中的双重过程。上述结果为分析具有相互作用的这种分支系统的大时空行为奠定了基础。特别是，我们在那里研究了长期行为和家族(或谱系)结构的普遍性，当我们从大的空间和时间尺度看的时候。
The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by a countable group, for example $Z^d$ or the hierarchical group. As type space we allow continua and describe populations in one colony as measures on the type space. The spatial components of the system interact via migration. Instead of the classical independence assumption on the evolution of different families of the branching population, we introduce interaction between the families through a state dependent branching rate of individuals and in addition state dependent mean offspring of individuals. However for most results we consider the critical case in this work. The systems considered arise as diffusion limits of critical multiple type branching random walks on a countable group with interaction between individual families induced by a branching rate and offspring mean for a single particle, which depends on the total population at the site at which the particle in question is located. The main purpose of this paper is to construct the measure valued diffusions in question, characterize them via well-posed martingale problems and finally determine their longtime behavior, which includes some new features. Furthermore we determine the dynamics of two functionals of the system, namely the process of total masses at the sites and the relative weights of the different types in the colonies as system of interacting diffusions respectively time-inhomogeneous Fleming-Viot processes. This requires a detailed analysis of path properties of the total mass processes. In addition to the above mentioned systems of interacting measure valued processes we construct the corresponding historical processes via well-posed martingale problems. Historical processes include information on the family structure, that is, the varying degrees of relationship between individuals. Ergodic theorems are proved in the critical case for both the process and the historical process as well as the corresponding total mass and relative weights functionals. The longtime behavior differs qualitatively in the cases in which the symmetrized motion is recurrent respectively transient. We see local extinction in one case and honest equilibria in the other. This whole program requires the development of some new techniques, which should be of interest in a wider context. Such tools are dual processes in randomly fluctuating medium with singularities and coupling for systems with multi-dimensional components. The results above are the basis for the analysis of the large space-time scale behavior of such branching systems with interaction carried out in a forthcoming paper. In particular we study there the universality properties of the longtime behavior and of the family (or genealogical) structure, when viewed on large space and time scales.