Conformal non-removability of the Brownian graph

Abstract

We prove that the graph of a standard one-dimensional Brownian motion, $Γ_B=\{(t,B_t):t \in [0,1]\}$, 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 $α\in (0,2/3)$, including the previously open range $α\in [1/2, 2/3)$. Together with the known removability result for $α>2/3$, this identifies $2/3$ as the threshold for conformal removability of Hölder graphs.