 Solution

A process $(X_t{)}_{t\ge 0}$ is Brownian motion in ${\mathbb{R}}^d$ when $X_0=0$, its paths are almost surely continuous, and for $0\le t_0<\cdots <t_k$ the increments $X_{t_j}-X_{t_{j-1}}$ are independent centered Gaussian vectors with covariance $(t_j-t_{j-1})I_d$.