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2019 Directed, cylindric and radial Brownian webs
David Coupier, Jean-François Marckert, Viet Chi Tran
Electron. J. Probab. 24: 1-48 (2019). DOI: 10.1214/18-EJP255

Abstract

The Brownian web (BW) is a collection of coalescing Brownian paths $(W_{(x,t)},(x,t) \in \mathbb{R} ^2)$ indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.

Citation

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David Coupier. Jean-François Marckert. Viet Chi Tran. "Directed, cylindric and radial Brownian webs." Electron. J. Probab. 24 1 - 48, 2019. https://doi.org/10.1214/18-EJP255

Information

Received: 26 July 2017; Accepted: 7 December 2018; Published: 2019
First available in Project Euclid: 21 March 2019

zbMATH: 07055658
MathSciNet: MR3933199
Digital Object Identifier: 10.1214/18-EJP255

Subjects:
Primary: 60D05, 60G52, 60J05, 60J65
Secondary: 60E99, 60G57

JOURNAL ARTICLE
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Vol.24 • 2019
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