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PreprintCC BY 4.0arXiv

Conformal non-removability of the Brownian graph

  • Haoyu Liu
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Abstract

We prove that the graph of a standard one-dimensional Brownian motion, ΓB={(t,Bt):t ∈ [0,1]}, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable α-Hölder graphs for every α∈ (0,2/3), including the previously open range α∈ [1/2, 2/3). Together with the known removability result for α>2/3, this identifies 2/3 as the threshold for conformal removability of Hölder graphs.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026conformal,
  title = {{Conformal non-removability of the Brownian graph}},
  author = {Haoyu Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00798},
  eprint = {2610.00798},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00798},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Conformal non-removability of the Brownian graph
AU  - Haoyu Liu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00798
UR  - https://arxiv.org/abs/2610.00798
AB  - We prove that the graph of a standard one-dimensional Brownian motion, Γ_B={(t,B_t):t ∈ [0,1]}, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable α-Hölder graphs for every α∈ (0,2/3), including the previously open range α∈ [1/2, 2/3). Together with the known removability result for α>2/3, this identifies 2/3 as the threshold for conformal removability of Hölder graphs.
ER  -