A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations

Oksendal, Bernt; Stroock, Daniel W.

Annales de l'Institut Fourier, Tome 32 (1982) no. 4, pp. 221-232.

La distribution de sortie d’un processus continu fortement markovien dans ${\mathbf{R}}^n$ est caractérisée comme une limite faible étoile de balayages sphériques de mesures, quand la première mesure est la mesure de Dirac. Ensuite on prouve que deux tels processus avec la même distribution de sortie des boules (commençant au centre) ont la même distribution de sortie de tous les ensembles ouverts, quand leur temps de sortie de boules sont p.s. finis. En conséquence, un processus continu fortement markovien dans ${\mathbf{R}}^n$ pour lequel les distributions de sortie sont préservées par les rotations, translations et dilatations, doit être un mouvement brownien moyennant un changement d’horaire. Aussi, nous obtenons une réciproque de la propriété moyenne des fonctions harmoniques.

The exit distribution for open sets of a path-continuous, strong Markov process in ${\mathbf{R}}^n$ is characterized as a weak star limit of successive spherical sweepings of measures, starting with the unit point mass. Then this is used to prove that two path-continuous strong Markov processes with identical exit distributions from balls when starting form the center, have identical exit distributions from all opens sets, provided they both exit a.s. from bounded sets. This implies that the only path-continuous, strong Markov process whose exit distributions are preserved by rotations, translations and dilatations is the Brownian motion, possibly with a changed time scale. For $n=2$ this is a converse of P. Lévy’s theorem about conformal invariance of Brownian motion. Finally we obtain a converse of the mean value property for harmonic functions.

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