On the Sobolev removability of the graph of one-dimensional Brownian motion

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