= Solution
On a <probability space> $(\Omega,\mathcal F,\mathbb P)$, a standard one-dimensional <Brownian motion> is a real-valued <stochastic process> $B=(B_t)_{t\geq0}$ with $B_0=0$ <almost surely>, continuous <sample paths> <almost surely>, and <independent increments>, such that
$$
\boxed{B_t-B_s\sim N(0,t-s)\qquad(0\leq s<t).}
$$
Here the second parameter is the <variance>. Thus its <Brownian increments> have the <normal distribution> with <mean> zero and <variance> equal to elapsed time. Equivalently, relative to a specified <filtration> $(\mathcal F_t)$, require $B$ to be an <adapted process> and $B_t-B_s$ to be <independent> of $\mathcal F_s$ with this <normal distribution>. Taking the <natural Brownian filtration> recovers the first definition.
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