编织、网络与流
Weaves, webs and flows
AI总结：
本文提出“编织”这一由非交叉路径覆盖时空的随机集合的一般理论，证明编织空间可划分为等价类，其中网络为极小元、流为极大元，并据此建立基于有限粒子运动弱收敛的一般弱收敛准则。
摘要：
我们引入“编织”（weaves）这一概念，它是由覆盖时空 $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$ 的非交叉 càdlàg 路径构成的随机集合。布朗网络（Brownian web）是编织的一个例子，但我们工作的一个关键特点是，我们不假设粒子运动具有任何特定分布。相反，我们提出了一个关于编织的结构、刻画和弱收敛的一般理论。我们证明，编织空间具有一种特别优美的几何结构，涉及划分为等价类，其中每个等价类包含一对被称为“网络”（web）和“流”（flow）的特殊对象。网络是布朗网络的天然推广，而流则提供了随机流的逐路径表示。此外，编织空间上存在一个自然的偏序，用以刻画路径覆盖时空的效率，在该偏序下，网络恰好是极小编织，而流恰好是极大编织。这一结构是建立一般编织弱收敛准则的关键，该准则基于有限粒子运动集合的弱收敛。
英文摘要：
We introduce weaves, which are random sets of non-crossing càdlàg paths that cover space-time $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$. The Brownian web is one example of a weave, but a key feature of our work is that we do not assume that particle motions have any particular distribution. Rather, we present a general theory of the structure, characterization and weak convergence of weaves. We show that the space of weaves has a particularly appealing geometry, involving a partition into equivalence classes under which each equivalence class contains a pair of distinguished objects known as a web and a flow. Webs are natural generalizations of the Brownian web and the flows provide pathwise representations of stochastic flows. Moreover, there is a natural partial order on the space of weaves, characterizing the efficiency with which paths cover space-time, under which webs are precisely minimal weaves and flows are precisely maximal weaves. This structure is key to establishing weak convergence criteria for general weaves, based on weak convergence of finite collections of particle motions.
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