= Conditional characteristic-function criterion for Brownian increments
{title2=$\mathbb E[e^{i\theta(X_t-X_s)}\mid\mathcal F_s]=e^{-\theta^2(t-s)/2}$}
For an adapted continuous process starting at zero, the displayed conditional <characteristic function> identifies every increment as $N(0,t-s)$ and makes it independent of the preceding sigma-field. Multiply by the indicator of an event in that sigma-field, then use the <uniqueness theorem for characteristic functions> for finite measures. Thus the process is a <Brownian motion> in the given filtration.
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