If $B$ and $f(B)$ are Brownian motions, then $f$ is affine

TL;DR

This paper proves that if a function transforms a Brownian motion into another Brownian motion without time change, then the function must be affine, and similarly characterizes solutions to both Laplace and eikonal equations.

Contribution

It establishes a rigidity result for functions preserving Brownian motion properties and characterizes solutions to two fundamental PDEs as affine functions.

Findings

Functions transforming Brownian motions are affine.

Only affine functions solve both Laplace and eikonal equations.

Provides a new characterization of affine functions via stochastic processes.

Abstract

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.