All Questions

There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them Denote by $E$ the Banach space ...

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 ...