= Lévy characterization of multidimensional Brownian motion
{c}
{title2=$[M^i,M^j]_t=\delta_{ij}t$}
A continuous vector-valued <local martingale> $M$ starting at zero is standard multidimensional <Brownian motion> if its <quadratic covariations> satisfy $[M^i,M^j]_t=\delta_{ij}t$. For each deterministic vector $\xi$, the <Itô formula> makes $\exp(i\xi\cdot M_t+|\xi|^2t/2)$ a complex <local martingale>. Its absolute value is bounded on each finite horizon, so it is a true <martingale>. Its <conditional expectation> identity gives
$$
\mathbb E[e^{i\xi\cdot(M_t-M_s)}\mid\mathcal F_s]=e^{-|\xi|^2(t-s)/2}.
$$
This conditional <characteristic function> identifies a centered <multivariate normal distribution> with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent <Brownian motions>. The converse follows immediately from the standard coordinate <quadratic covariations>.
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