= Solution
A standard one-dimensional <Brownian motion> is a real-valued <stochastic process> $(B_t)_{t\geq0}$ satisfying $B_0=0$ <almost surely>, having <continuous paths> <almost surely>, and having <independent increments> with
$$
\boxed{B_t-B_s\sim N(0,t-s)\qquad(0\leq s<t).}
$$
Here $N(0,t-s)$ is the <normal distribution> with zero <expected value> and <variance> $t-s$. Thus the increments are also <stationary increments>.
If <Brownian motion> is defined relative to a specified <filtration> $(\mathcal F_t)$, additionally require that it is an <adapted process> and that $B_t-B_s$ is <independence>[independent] of $\mathcal F_s$. For its <natural filtration>, this follows from <independent increments>. \b[These are the defining properties of standard <Brownian motion>.]
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