Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

Abstract

By the Cameron--Martin theorem, if a function $f$ is in the Dirichlet space $D$, then $B+f$ has the same a.s. properties as standard Brownian motion, $B$. In this paper we examine properties of $B+f$ when $f \notin D$. We start by establishing a general 0-1 law, which in particular implies that for any fixed $f$, the Hausdorff dimension of the image and the graph of $B+f$ are constants a.s. (This 0-1 law applies to any L\'evy process.) Then we show that if the function $f$ is H\"older$(1/2)$, then $B+f$ is intersection equivalent to $B$. Moreover, $B+f$ has double points a.s. in dimensions $d\le 3$, while in $d\ge 4$ it does not. We also give examples of functions which are H\"older with exponent less than $1/2$, that yield double points in dimensions greater than 4. Finally, we show that for $d \ge 2$, the Hausdorff dimension of the image of $B+f$ is a.s. at least the maximum of 2 and the dimension of the image of $f$.