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The Brownian web as a random \(\mathbb{R} \)-tree. (English) Zbl 07733587

This paper by Giuseppe Cannizzaro and Martin Hairer presents a construction of the Brownian web as a random variable in the space of spatial \(\mathbb{R} \)-trees that are coded by a continuous function. The Brownian web is an uncountable collection of one-dimensional coalescing Brownian motions, starting from every point in space and time \(\mathbb{R}^2\) simultaneously, where \(\mathbb{R} \) denotes the set of real numbers. This construction provides a stronger topology of the \(\mathbb{R} \)-trees than the classical one, making it a complete separable metric space. This also allows for more continuous functions of the Brownian web and eliminates potential pathological behaviors. In the context of spatial \(\mathbb{R} \)-trees, the paper determines properties of the Brownian web, such as its box-counting dimension, and recovers earlier ones, including duality, special points, and convergence of graphical representations of coalescing random walks.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60D05 Geometric probability and stochastic geometry

References:

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