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

Abstract

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

21 pages, 3 figures