= Solution
A standard real <Brownian motion> relative to a <filtration> $(\mathcal F_t)$ is an <adapted> <stochastic process> with $B_0=0$ almost surely, continuous paths almost surely, and
$$
\boxed{B_t-B_s\sim\mathcal N(0,t-s),\qquad B_t-B_s\text{ independent of }\mathcal F_s\quad(0\leq s<t).}
$$
Equivalently, it has <independent increments> and <stationary increments>, each with a <normal distribution> of mean zero and <variance> equal to the elapsed time, together with continuous paths and the stated initial value. The filtration may be its <natural filtration>. A <Brownian motion> started at $z$ is $z+W_t$ for a standard process $W$; multidimensional standard <Brownian motion> has independent standard real coordinate processes. \b[The normalization is <variance> $t$ at time $t$.]
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