February 2022 Minkowski content of Brownian cut points
Nina Holden, Gregory F. Lawler, Xinyi Li, Xin Sun
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(1): 455-488 (February 2022). DOI: 10.1214/21-AIHP1159


Let W(t), 0tT, be a Brownian motion in Rd, d=2,3. We say that x is a cut point for W if x=W(t) for some t(0,T) such that W[0,t) and W(t,T] are disjoint. In this work, we prove that a.s. the Minkowski content of the set of cut points for W exists and is finite and non-trivial.

Soit W(t), 0tT un mouvement brownien sur Rd, d=2,3. On dit que x est un point de coupure pour W si x=W(t) pour un certain t(0,T) tel que W[0,t) et W(t,T] sont disjoints. Dans cet article, nous montrons que le contenu de Minkowski de l’ensemble des points de coupure pour W existe p.s., et qu’il est p.s. fini et non trivial.

Funding Statement

The first author was supported in part by a fellowship from the Research Council of Norway and partially supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
The second author was supported by NSF grant DMS-1513036.
The third author was supported by NSFC (No. 12071012) and the National Key R&D Program of China (No. 2020YFA0712900).
The fourth author was supported by a Junior Fellow award from the Simons Foundation, and NSF Grant DMS-1811092 and DMS-2027986.


The authors would like to thank Dapeng Zhan for comments and suggestions on a previous version of the draft, especially regarding the cut-point Green’s function. We also want to thank the referee for careful reading of the paper and for numerous helpful suggestions.


Download Citation

Nina Holden. Gregory F. Lawler. Xinyi Li. Xin Sun. "Minkowski content of Brownian cut points." Ann. Inst. H. Poincaré Probab. Statist. 58 (1) 455 - 488, February 2022. https://doi.org/10.1214/21-AIHP1159


Received: 29 October 2018; Revised: 25 January 2021; Accepted: 2 February 2021; Published: February 2022
First available in Project Euclid: 2 February 2022

MathSciNet: MR4374682
zbMATH: 1487.60148
Digital Object Identifier: 10.1214/21-AIHP1159

Primary: 60J65 , 82B27

Keywords: Brownian motion , cut point , Minkowski content

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


This article is only available to subscribers.
It is not available for individual sale.

Vol.58 • No. 1 • February 2022
Back to Top