If B and f(B) are Brownian motions, then f is affine
classification
🧮 math.PR
keywords
affinebrownianequationmotionsthenby-productchangeeikonal
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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 a by-product of the proof, it is shown that the only functions which are solutions to both the Laplace equation and the eikonal equation are affine.
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