Solution
= Solution
A scalar process is a <Brownian motion> in $(\mathcal F_t)$ when it is adapted, has continuous sample paths, starts at zero almost surely, and, for every $0\leq s<t$,
$$
\boxed{X_t-X_s\sim N(0,t-s),\qquad
X_t-X_s\text{ is independent of }\mathcal F_s.}
$$
Independence from the whole past sigma-field is the essential filtration requirement. It implies independent increments over disjoint intervals; the variance $t-s$ fixes the standard Brownian time scale.