Brownian Motion as a Gaussian Process

Key fact

For any

$$ 0the increments

$$ B_{t_1},\quad B_{t_2}-B_{t_1},\quad \dots,\quad B_{t_k}-B_{t_{k-1}} $$

are independent normal random variables.

The vector

$$ (B_{t_1},\dots,B_{t_k}) $$

is multivariate normal for every finite set of times. Therefore Brownian motion is a Gaussian process.

For $s\le t$,

$$ \mathrm{Cov}(B_s,B_t)=s. $$

Equivalently,

$$ \mathrm{Cov}(B_s,B_t)=\min(s,t). $$