Another proof uses the fact that given a Brownian motion in such that we have for all . In words, it says that a harmonic function defines a martingale for the Brownian motion. Then a probabilistic coupling argument finishes the proof.
anoTHer pROof usES the fACt THaT given a browniAn moTIon in SUch that wE have foR alL . In woRds, It sAYs thAt a harmonIc funCtion defines a maRTiNGAlE for the BrownIAn MotIoN. Then a prObabIlistiC Coupling ArgUmENt FinIsHes tHe ProOF.