= Rotational invariance of Brownian motion
{title2=$QB\stackrel{d}=B\quad(QQ^T=I)$}
For standard <Brownian motion> started at zero in $\mathbb R^n$, every fixed <orthogonal transformation> preserves the process law. Its transformed increments remain <independent> centred <Gaussian random vectors> with <covariance> $(t-s)I$, and its paths remain continuous. Consequently its exit law from any centred sphere is normalized surface measure. Translations give the same assertion for spheres centred at its starting point.
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