Certain local properties of the Wiener process $W_t$ are quick to prove at $t = 0$, for instance:
- almost surely $W_t$ is monotonous on no interval beginning at $t = 0$;
- almost surely $W_t$ is not right-differentiable at $t = 0$.
Then, because $W_{t_0+t} - W_t$ is a Wiener process, we can conclude that
- for any $t \ge 0$, almost surely $W_t$ is monotonous on no interval beginning at $t$,
- for any $t \ge 0$, almost surely $W_t$ is not right-differentiable at $t$.
Thus at any countable set of times, in particular a dense one, both hold almost surely. Now, if the first holds at a certain $t = t_0$ then it holds for all $t$ in a nonempty open interval to the right of $t_0$, so almost surely it must hold for all $t$. We have successfully exchanged “for all $t$” and “almost surely” for the first property.
For the second property we are less fortunate: there are continuous functions not differentiable on a countable dense subset but differentiable on its complement. Yet this “exchange of quantifiers” is still valid here: almost surely $W_t$ is nowhere differentiable. However the proofs I have seen are considerably more involved than that for only $t = 0$.
Hence my question: is there a clean condition on the events $A_t$ under which $$\inf_{t\ge0} \Pr(W \in A_t) = 1 \implies \Pr\Bigl( W \in \bigcap_{t\ge0} A_t \Bigr) = 1 \text?$$ I am interested in “local” events $A_t$; let's suppose $$A_t \in \bigcap_{\varepsilon>0} \sigma(\, W_{t+\delta} \colon 0 \le \delta < \varepsilon \,) \text, \quad \text{$t \ge 0$.}$$ An obvious starting point would be $$A_t \subseteq \bigcup_{\varepsilon>0} \bigcap_{0\le\delta<\varepsilon} A_{t+\delta} \text,$$ as with the first property concerning monotonicity above. It would be extra nice if the condition applied to the second property about differentiability.
