A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations
Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 221-232.

The exit distribution for open sets of a path-continuous, strong Markov process in 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.

La distribution de sortie d’un processus continu fortement markovien dans 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 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.

@article{AIF_1982__32_4_221_0,
     author = {Oksendal, Bernt and Stroock, Daniel W.},
     title = {A characterization of harmonic measure and {Markov} processes whose hitting distributions are preserved by rotations translations and dilatations},
     journal = {Annales de l'Institut Fourier},
     pages = {221--232},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {32},
     number = {4},
     year = {1982},
     doi = {10.5802/aif.901},
     mrnumber = {84g:60125},
     zbl = {0489.60078},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.901/}
}
TY  - JOUR
TI  - A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations
JO  - Annales de l'Institut Fourier
PY  - 1982
DA  - 1982///
SP  - 221
EP  - 232
VL  - 32
IS  - 4
PB  - Imprimerie Durand
PP  - 28 - Luisant
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.901/
UR  - https://www.ams.org/mathscinet-getitem?mr=84g:60125
UR  - https://zbmath.org/?q=an%3A0489.60078
UR  - https://doi.org/10.5802/aif.901
DO  - 10.5802/aif.901
LA  - en
ID  - AIF_1982__32_4_221_0
ER  - 
%0 Journal Article
%T A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations
%J Annales de l'Institut Fourier
%D 1982
%P 221-232
%V 32
%N 4
%I Imprimerie Durand
%C 28 - Luisant
%U https://doi.org/10.5802/aif.901
%R 10.5802/aif.901
%G en
%F AIF_1982__32_4_221_0
Oksendal, Bernt; Stroock, Daniel W. A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations. Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 221-232. doi : 10.5802/aif.901. https://aif.centre-mersenne.org/articles/10.5802/aif.901/

[1] J. R. Baxter, Harmonic functions and mass cancellation, Trans. Amer. Math. Soc., 245 (1978), 375-384. | MR: 80a:60085 | Zbl: 0391.60065

[2] A. Bernard, E. A. Campbell and A. M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier, 29, 1 (1979), 207-228. | Numdam | MR: 81b:30088 | Zbl: 0386.30029

[3] H. M. Blumenthal, R. K. Getoor and H. P. Mckean, Jr., Markov processes with identical hitting distributions, Illinois Journal of Math., 6 (1962), 402-420. | MR: 25 #5550 | Zbl: 0133.40903

[4] H. M. Blumenthal, R.K. Getoor and H. P. Mckean, Jr., A supplement to "Markov processes with identical hitting distributions", Illinois Journal of Math., 7 (1963), 540-542. | MR: 27 #3018 | Zbl: 0211.48602

[5] T. W. Gamelin and H. Rossi, Jensen measures, In Birtel (ed.) : Function Algebras, Scott, Foresman and Co. (1966).

[6] D. Heath, Functions possessing restricted mean value properties, Proc. Amer. Math. Soc., 41 (1973), 588-595. | MR: 48 #11538 | Zbl: 0251.31004

[7] O. D. Kellogg, Converses of Gauss' theorem on the arithmetic mean, Trans. Amer. Math. Soc., 36 (1934), 227-242. | MR: 1501739 | Zbl: 0009.11205

[8] H. P. Mckean, Jr., Stochastic Integrals, Academic Press, 1969. | MR: 40 #947 | Zbl: 0191.46603

[9] S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, 1968.

[10] M. Rao, Brownian Motion and Classical Potential Theory, Lecture Notes Series No 47, Aarhus Universitet, 1977. | MR: 55 #13589 | Zbl: 0345.31001

[11] W. A. Veech, A zero-one law for a class of random walks and a converse to Gauss' mean value theorem, Annals of Math., 97 (1973), 189-216. | MR: 46 #9370 | Zbl: 0282.60048

[12] W. A. Veech, A converse to the mean value theorem for harmonic functions, Amer. J. Math., 97 (1975), 1007-1027. | MR: 52 #14330 | Zbl: 0324.31002

Cited by Sources: