  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 $f$.
