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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?

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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.

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