Suppose that $B$ is a one-dimensional Brownian motion and let $Γ= \{ (t, B_t) : t \in [0,1]\}$ be the graph of $B|_{[0,1]}$. We characterize the Sobolev removability properties of $Γ$ by showing that $Γ$ is almost surely not $W^{1,p}$--removable for all $p \in [1, \infty)$ but is almost surely $W^{1,\infty}$--removable.
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