The Brownian Web as a random $\mathbb R$-tree

TL;DR

This paper constructs the Brownian Web as a random $ ext{\mathbb R}$-tree within a stronger topology, enabling more continuous analysis and excluding pathological behaviors, and explores its properties and characterizations.

Contribution

It provides a novel construction of the Brownian Web as a random $ ext{\mathbb R}$-tree with a stronger topology, enhancing analytical continuity and understanding.

Findings

Constructed the Brownian Web as a random $ ext{\mathbb R}$-tree.

Introduced a modified topology making the space complete and separable.

Analyzed properties like box-counting dimension, duality, and convergence behaviors.

Abstract

Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\mathbb R^2$, as a random variable taking values in the space of (spatial) $\mathbb R$-trees. This gives a stronger topology than the classical one {(i.e.\ Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g.\ its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.