A Wiener test for integrals of brownian motion and the existence of smooth curves in nowhere dense sets

AbstractLet Jωx(t) = x + ∝0t bω(s) ds, where bω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process Jωx(t): Let K be a compact plane set and let x ϵ K. Then if ∑ 2nM1(An(x)⧹K) < ∞ (where An(x) = {2−n−1 ⩽ ¦ z − x ¦ ⩽ 2−n} and M1 denotes one-dimensional Hausdorff content), the process Jωx(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes