Weaves, webs and flows

TL;DR

This paper develops a general theory for 'weaves', random non-crossing path sets covering space-time, unifying structures like the Brownian web and stochastic flows without assuming specific particle motion distributions.

Contribution

It introduces a comprehensive framework for analyzing weaves, including their structure, characterization, and convergence, generalizing previous models like the Brownian web.

Findings

The space of weaves has a rich geometric structure with distinguished objects called webs and flows.

Webs are minimal and generalize the Brownian web; flows are maximal and represent stochastic flows.

The theory provides weak convergence criteria based on finite particle collections.

Abstract

We introduce weaves, which are random sets of non-crossing c\`{a}dl\`{a}g 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.