What's the difference between
"A stochastic process has infinitely many zeros in every interval almost surely."
and
"A stochastic process has infinitely many zeros in every interval."
?
In my opinion there is no difference since an interval is no P-nullset. I'm trying to imagine paths of Brownian motions. But if the two statements above mean the same you couldn't image any Brownian path. But it's not easy for me since most properties are almost surely. Can anyone help me?
'... an interval is no P-null set...': You are confusing the Lebesgue measure on the time line with the underlying probability measure.
A real-valued stochastic process is a measurable mapping from a probability space, $(\Omega,\mathcal{F},P)$ to the set of real-valued functions equipped with a proper sigma algebra.
The first statement means this: If you collect every $\omega \in \Omega$ such that the function $X(\omega)$ has infinitely many zeros in every interval, the resulting set is an element of $\mathcal{F}$ and that it has measure $1$.
The second statement means this: For every $\omega \in \Omega$, the function $X(\omega)$ has infinitely many zeros in every interval.