We define

$p_T(f):= \mathbb{P}(\forall_{s\leq T}B_s \geq f(s)-1),$

where $B$ is a Brownian motion such that $B_0 =0$ and $f:\mathbb{R}_+ \mapsto \mathbb{R}$ is some (continuous) function. I am interested in understanding the random variable:

$X_T=p_T(\tilde{B}),$

where again $\tilde{B}$ is a Brownian motion (I use $\tilde{B}$ just to avoid any confusion). What is easy

$\mathbb{E}X = \mathbb{P}(\forall_{s\leq T}B_s \geq \tilde{B}_s-1) \sim T^{-1/2}.$

I would be interested in knowing if e.g.

$\mathbb{P}(X_T \geq T^{-A})\rightarrow 1,$

for $A$ large enough.