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

Directed, cylindric and radial Brownian webs. (English) Zbl 1456.60206

Electron. J. Probab. 24, Paper No. 20, 48 p. (2019).
Summary: 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 C.F. Coletti and L.A. Valencia [“Scaling limit for a family of random paths with radial behavior”, Preprint, arXiv:1310.6929] is shown to converge to the CBW.

Cited in 2 Documents

MSC:

60J65 Brownian motion
60D05 Geometric probability and stochastic geometry
60J05 Discrete-time Markov processes on general state spaces
05C80 Random graphs (graph-theoretic aspects)
60G52 Stable stochastic processes
60G57 Random measures

Keywords:

Brownian web; navigation algorithm; random spanning forests; weak convergence of stochastic processes
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\textit{D. Coupier} et al., Electron. J. Probab. 24, Paper No. 20, 48 p. (2019; Zbl 1456.60206)
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