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

Abstract

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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.

How to cite

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Oksendal, 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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  3. [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. Zbl0133.40903MR25 #5550
  4. [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. Zbl0211.48602MR27 #3018
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  7. [7] O. D. KELLOGG, Converses of Gauss' theorem on the arithmetic mean, Trans. Amer. Math. Soc., 36 (1934), 227-242. Zbl0009.11205MR1501739
  8. [8] H. P. MCKEAN, Jr., Stochastic Integrals, Academic Press, 1969. Zbl0191.46603MR40 #947
  9. [9] S. C. PORT and C. J. STONE, Brownian Motion and Classical Potential Theory, Academic Press, 1968. Zbl0413.60067
  10. [10] M. RAO, Brownian Motion and Classical Potential Theory, Lecture Notes Series No 47, Aarhus Universitet, 1977. Zbl0345.31001MR55 #13589
  11. [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. Zbl0282.60048MR46 #9370
  12. [12] W. A. VEECH, A converse to the mean value theorem for harmonic functions, Amer. J. Math., 97 (1975), 1007-1027. Zbl0324.31002MR52 #14330

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