Abstract
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 simultaneously, as a random variable taking values in a space of (spatial) -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 -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.
Acknowledgments
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.
Citation
Giuseppe Cannizzaro. Martin Hairer. "The Brownian Web as a random -tree." Electron. J. Probab. 28 1 - 47, 2023. https://doi.org/10.1214/23-EJP984
