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

Bernt Oksendal; Daniel W. Stroock

Annales de l'institut Fourier (1982)

- Volume: 32, Issue: 4, page 221-232
- ISSN: 0373-0956

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topOksendal, Bernt, and 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 32.4 (1982): 221-232. <http://eudml.org/doc/74562>.

@article{Oksendal1982,

abstract = {The exit distribution for open sets of a path-continuous, strong Markov process in $\{\bf R\}^n$ 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.},

author = {Oksendal, Bernt, Stroock, Daniel W.},

journal = {Annales de l'institut Fourier},

keywords = {spherical sweepings of measures; harmonic functions},

language = {eng},

number = {4},

pages = {221-232},

publisher = {Association des Annales de l'Institut Fourier},

title = {A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations},

url = {http://eudml.org/doc/74562},

volume = {32},

year = {1982},

}

TY - JOUR

AU - Oksendal, Bernt

AU - Stroock, Daniel W.

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

PB - Association des Annales de l'Institut Fourier

VL - 32

IS - 4

SP - 221

EP - 232

AB - The exit distribution for open sets of a path-continuous, strong Markov process in ${\bf R}^n$ 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 - eng

KW - spherical sweepings of measures; harmonic functions

UR - http://eudml.org/doc/74562

ER -

## References

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