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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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