Directed, cylindric and radial Brownian webs

TL;DR

This paper introduces the cylindric Brownian web (CBW) as an interpolation between directed and radial Brownian webs, establishing its properties, duality, and convergence from discrete models, thus advancing the understanding of coalescing Brownian paths on cylindrical structures.

Contribution

The paper defines the CBW and RBW, analyzes their properties, and proves convergence of discrete radial forests to these continuous structures, bridging directed and radial models.

Findings

CBW has exponential coalescence times and a unique bi-infinite path.

The dual of CBW contains a pair of reflected Brownian motions.

Discrete radial forests converge to the RBW under suitable rescaling.

Abstract

The Brownian web (BW) is a collection of coalescing Brownian paths 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.