Isolated Zeros For Brownian Motion With Variable Drift

It is well known that standard one-dimensional Brownian motion B(t) has no isolated zeros almost surely. We show that for any α < 1/2 there are α-Hölder continuous functions f for which the process B −f has isolated zeros with positive probability. We also prove that for any continuous function f, the zero set of B − f has Hausdorff dimension at least 1/2 with positive probability, and 1/2 is an upper bound on the Hausdorff dimension if f is 1/2-Hölder continuous or of bounded variation.