Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$?
I actually need to estimate these probabilities (even more precisely, just determine whether they are zero or positive) for a different process, governed by a certain SDE with a drift and values only on $[0,1]$, but this would be a good start. Obviously, $\mathbb{P}[W_t=y]=0$ for any fixed $t>0$. But I'm not quite sure how to move from here to the whole path. The Wiener process should visit any point infinitely often, but the questions is whether the infinity here is big enough to make the probability positive.
Thanks for comments, hints or links to relevant books/articles.