Let , , be a Brownian motion in , . We say that x is a cut point for W if for some such that and 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 , un mouvement brownien sur , . On dit que x est un point de coupure pour W si pour un certain tel que et 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.
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.
"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