Let $x\colon[0,1]\to \mathbb{R}$ be a continuous path with, and let $B_t$ be a standard Brownian motion on some probability space.
I want to compute the following conditional probability:
$$ \mathbb{E}\left[f(B(t))\,\Big|\quad \sup_{s\in[0,t]}|B(s)-x(s)|<\varepsilon\right]. $$ Where $f$ is a given function.
Does this make sense?
I would like to know if it depends on the path of $x$, or only on $x(t)$? Is it just $f(x(t))$?