Gaussian Process Characterization of Standard Brownian Motion

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $B=(B_t)_{t\ge0}$ be a stochastic process on $(\Omega,\mathcal{F},P)$ indexed by the nonnegative real numbers, $\mathbb{R}$ being the real numbers. For real numbers $s$ and $t$, let $\min(s,t)$ denote the smaller of $s$ and $t$.

Then $B$ is a standard Brownian motion if and only if the following two conditions hold:

(a) $B$ is a Gaussian process, and its expectations and covariances of square-integrable random variables — defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector — satisfy

(b) almost surely, the path

is continuous at every point of $[0,\infty)$ relative to $[0,\infty)$, both $[0,\infty)$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

No separate initial-value condition is imposed: condition (a) alone implies $P(B_0=0)=1$, which is the almost sure initial-value clause of Standard Brownian Motion.