A stronger topology for the Brownian web

TL;DR

This paper introduces a new topology for the Brownian web, enabling direct proofs of convergence for various models' distributions to their Brownian web counterparts, enhancing analytical tools in stochastic processes.

Contribution

It develops a stronger topology for the Brownian web, facilitating direct convergence proofs for models related to coalescing random walks and their applications.

Findings

Proved convergence of voter model persistence probability to the Brownian web

Established convergence of weight distribution in silo models to the Brownian web

Applied the topology to river basin model distributions

Abstract

We propose a metric space of coalescing pairs of paths on which we are able to prove (more or less) directly convergence of objects such as the persistence probability in the (one dimensional, nearest neighbor, symmetric) voter model or the diffusively rescaled weight distribution in a silo model (as well as the equivalent output distribution in a river basin model), interpreted in terms of (dual) diffusively rescaled coalescing random walks, to corresponding objects defined in terms of the Brownian web.