If $$B$$ and $$f(B)$$ are Brownian motions, then $$f$$ is affine.(English)Zbl 1369.31011

Summary: 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.

MSC:

 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 35Q60 PDEs in connection with optics and electromagnetic theory 60J65 Brownian motion
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