Open Access
2023 The Brownian Web as a random R-tree
Giuseppe Cannizzaro, Martin Hairer
Author Affiliations +
Electron. J. Probab. 28: 1-47 (2023). DOI: 10.1214/23-EJP984


Motivated by [CH23], we provide a construction of the Brownian Web [TW98, FINR04], i.e. a family of coalescing Brownian motions starting from every point in R2 simultaneously, as a random variable taking values in a space of (spatial) 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 R-trees in [DLG05, BCK17] 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.

Funding Statement

Supported by the Institute of Mathematical Statistics (IMS) and the Bernoulli Society. GC gratefully acknowledges financial support via the EPSRC grant EP/S012524/1 and the UKRI FL Fellowship MR/W008246/1. MH gratefully acknowledges financial support from the Leverhulme trust via a Leadership Award, the ERC via the consolidator grant 615897:CRITICAL, and the Royal Society via a research professorship.


The authors would like to thank the referees for their very detailed reports which helped to improve the presentation of the paper. In particular, we are grateful to the referee who suggested the formulation of Theorem 1.1 as stated. GC would like to thank the Hausdorff Institute in Bonn for the kind hospitality during the programme “Randomness, PDEs and Nonlinear Fluctuations”, where part of this work was carried out.


Download Citation

Giuseppe Cannizzaro. Martin Hairer. "The Brownian Web as a random R-tree." Electron. J. Probab. 28 1 - 47, 2023.


Received: 9 February 2021; Accepted: 27 June 2023; Published: 2023
First available in Project Euclid: 1 August 2023

MathSciNet: MR4622477
zbMATH: 07733587
Digital Object Identifier: 10.1214/23-EJP984

Primary: 60G

Keywords: Brownian Castle , Brownian web , Random trees

Vol.28 • 2023
Back to Top