Non-existence of several random fractals in Brownian motion and Brownian loop soup

Yifan Gao; Xinyi Li; Runsheng Liu; Wei Qian

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arXiv · 2507.20324

Non-existence of several random fractals in Brownian motion and Brownian loop soup

Yifan Gao·Xinyi Li·Runsheng Liu·Wei Qian

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Abstract

We develop a unified approach to establish the non-existence of three types of random fractals: (1) the pioneer triple points of the planar Brownian motion, answering an open question in [7], (2) the pioneer double cut points of the planar and three-dimensional Brownian motions, and (3) the double points on the boundaries of the clusters of the planar Brownian loop soup at the critical intensity, answering an open question in [39]. These fractals have the common feature that they are associated with an intersection or disconnection exponent which yields a Hausdorff dimension ``exactly zero''.

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BibTeX RIS

Yifan Gao, Xinyi Li, Runsheng Liu, Wei Qian. 2025-07-27. Non-existence of several random fractals in Brownian motion and Brownian loop soup. https://arxiv.org/abs/2507.20324

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