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It is shown that if the processes $B$ and $f(B)$ are both Brownian motions(without a random time change) then $f$ must be an affine function. As aby-product of the proof, it is shown that the only functions which aresolutions to both the Laplace equation and the eikonal equation are affine.

If $B$ and $f(B)$ are Brownian motions, then $f$ is affine