Suppose $B_t$ is a standard Brownian motion defined on the probability space $(\Omega ,\;{\mathcal {F}},\;P) $ with a filtration and s.t. $\Omega$ is large enough for $(B_t)_{t\in\mathbb R_+}$ to exist on $\Omega$. If one wants to be more explicit one could take $\Omega =C(\mathbb R_+,\mathbb R)$.
I am looking for results regarding the probability of sample paths to belong to certain open balls in function space.
A sample path is, for some $\omega$ in $\Omega$, the function $B(\omega):\mathbb R_+\to\mathbb R$, $t\mapsto B_t(\omega)$, where we let $t \in [0,T]$ for a certain $T \in R$.
Specifically letting $B_1(\omega)$ be a specific realization of a Brownian motion process and take a fixed $\epsilon> 0$ what is known about
$$P(B(w) \in \mathcal{B}(B_1(w), \epsilon) ) ?$$
where I have denoted with $\mathcal{B}(B_1(w), \epsilon)$ the ball given by the supremum norm in function space,
${\displaystyle \|B(w)\|_{\infty }=\sup \left\{\,\left|B(\omega,t)\right|:t\in [0,T]\,\right\}.} $
One can also reformulate this question as: with what probability is the brownian motion going to be in an epsilon neighborhood of one of its possible sample paths?
This is the problem of the maximum of BM in compact intervals. Its distribution is well known, and follows easily from the so-called reflection principle and strong Markov property.