If we consider the process $$ Y^f_t := \sup_{s\leq t} \vert B_s \vert - f(s)$$ is there anything known about the distribution or at least the probability $\Bbb P (Y^f_t \leq 0 )$ ?
Of particular interest would be continuous functions $f$, and functions of the form $f_1 \equiv c$, where $ c>0 $ or $f_2 = c + \delta\cdot \Bbb 1_{[u,s)}$ where $s>u > 0 $.
For example one could ask if $\Bbb P (Y_t^{f_1} \leq 0) < \Bbb P (Y_t^{f_2} \leq 0)$, which is very reasonable at first sight but I see no easy way to prove it.