Solution
= Solution
An $\mathbb R^d$-valued process $(X_t)_{t\geq0}$ is a <Brownian motion> started at $x$ when:
* $X_0=x$ almost surely;
* its sample paths are almost surely continuous;
* increments over disjoint time intervals are independent; and
* for $0\leq s<t$, $X_t-X_s$ has the centered <multivariate normal distribution> with covariance matrix $(t-s)I_d$.