Electronic Journal of Probability

Directed, cylindric and radial Brownian webs

David Coupier, Jean-François Marckert, and Viet Chi Tran

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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.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 20, 48 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1553133829

Digital Object Identifier
doi:10.1214/18-EJP255

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 60G52: Stable processes 60J65: Brownian motion [See also 58J65] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G57: Random measures 60E99: None of the above, but in this section

Keywords
Brownian web navigation algorithm random spanning forests weak convergence of stochastic processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Coupier, David; Marckert, Jean-François; Chi Tran, Viet. Directed, cylindric and radial Brownian webs. Electron. J. Probab. 24 (2019), paper no. 20, 48 pp. doi:10.1214/18-EJP255. https://projecteuclid.org/euclid.ejp/1553133829


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