IF B AND f (B) ARE BROWNIAN MOTIONS, THEN f IS AFFINE

摘要

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.