What defines a standard Brownian motion

Give a definition of Brownian motion.

A continuous Gaussian process with independent, stationary increments. A standard Brownian motion $W$ has $W_0=0$, continuous paths, and for $s<t$ the increment $W_t-W_s$ is $N(0,t-s)$ and independent of the past. Useful consequences: $\operatorname{Cov}(W_s,W_t)=\min(s,t)$, quadratic variation $t$, paths nowhere differentiable, and it is the scaling limit of a random walk.