Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$).

Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$, $$ W_t=W_s\iff V_{f(t)}=V_{f(s)}. $$ Does $C$ have probability 0?

Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$).

Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$, $$ W_t=W_s\iff V_{f(t)}=V_{f(s)}. $$ Does $C$ have probability 0?

(The question arose in connection with a question by Noah Schweber.)