Tehranchi, Michael R. If \(B\) and \(f(B)\) are Brownian motions, then \(f\) is affine. (English) Zbl 1369.31011 Rocky Mt. J. Math. 47, No. 3, 947-953 (2017). 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. Cited in 1 Document MSC: 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 35Q60 PDEs in connection with optics and electromagnetic theory 60J65 Brownian motion Keywords:Brownian motion; harmonic function; Laplace equation; eikonal equation PDF BibTeX XML Cite \textit{M. R. Tehranchi}, Rocky Mt. J. Math. 47, No. 3, 947--953 (2017; Zbl 1369.31011) Full Text: DOI arXiv Euclid OpenURL