= Gaussian-process characterization of Brownian motion
{c}
{title2=$\mathbb E[B_sB_t]=\min(s,t)$}
A centered <Gaussian process> with <almost surely> continuous <sample paths> and <covariance> $\mathbb E[B_sB_t]=\min(s,t)$ is standard <Brownian motion>. Here <Gaussian process> means that every <finite-dimensional distribution> is a <multivariate normal distribution>; having separate <normal distributions> at each time would not suffice. This characterization is stated in https://www.stat.berkeley.edu/~pitman/s205s03/lecture15.pdf[Pitman, Basic Properties of Brownian Motion, Definition 15.2].
To compute the <Brownian covariance kernel> directly, suppose $0\leq s\leq t$. Since $B_t=B_s+(B_t-B_s)$ and the added <Brownian increment> is centered and <independent> of $B_s$,
$$
\operatorname{Cov}(B_s,B_t)=\mathbb E B_s^2+\mathbb E[B_s(B_t-B_s)]=s+0=s.
$$
Symmetry gives $\min(s,t)$ for arbitrary nonnegative times. For instance, $\operatorname{Cov}(B_1,B_4)=1$ and their <correlation> is $1/\sqrt{1\cdot4}=1/2$. Positions are therefore related even though disjoint <Brownian increments> are <independent>: both positions contain the displacement accumulated during the first unit of time. Conversely, taking differences of $\min(s,t)$ gives zero <covariance> for disjoint increments; their joint <multivariate normal distribution> turns this into <independence>.
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