 Solution

A standard Brownian motion is a real process $(B_t{)}_{t\ge 0}$ with $B_0=0$ almost surely, continuous paths, independent increments, and $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$. If a filtration is specified, require adaptedness and independence of each increment from the past sigma-algebra. Thus the increments are centered Gaussian with variance equal to elapsed time.