Summary
I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is is positive at $t=0$, that it is bounded from left by $0$ and by right by some $T>0$, that it is continuous, or even differentiable, and many other nice curvish features that may help making this question more feasible.
Formalizing
Let $\{B(t)\mid t\ge 0\}$ be a standard linear Brownian motion, and let $f:[0,T]\to\mathbb{R}$ be infinitely-differentiable (in $(0,T)$) real function with $T>0$ and $f(0)>0$. Let $A_f$ be the event "$\exists t\in(0,T]):\ B(t)=f(t)$", that is, the Brownian motion "hits" the graph of the function $f$.
The question is as follows: given $f$, what is $\mathbb{P}\left(A_f\right)$?
Attempt
All I could do is solve this for $f\equiv c>0$; in that case, if we define $M(t)=\max\{B(s)\mid 0\le s\le t\}$ we have that $$\mathbb{P}\left(A_f\right)=\mathbb{P}\left(M(T)\ge c\right)$$ and by reflection principle, the last probability equals $$2\mathbb{P}\left(B(T)\ge c\right)$$ and that can be solved using straight-forward normal cdf.
However, even for non-constant linear $f$'s that trick won't do; and $f(x)=1/x$ (with something at $0$, bounded from the right by some $T$) seems much harder. This is where I stop and post a question.