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Harmonic measure on 3-dimensional Brownian paths

Published online by Cambridge University Press:  24 October 2008

Krzysztof Burdzy
Krzysztof Burdzy
Affiliation:
University of Washington, Seattle W A 98195, U.S.A.
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Abstract

It is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 407 - 412
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Burdzy, K.. Geometric properties of the complement of 2-dimensional Brownian path. (Preprint.)Google Scholar
[2]Davis, B. and Salisbury, T.. Connecting Brownian paths. Ann. Probab., to appear.Google Scholar
[3]Doob, J. L.. Classical Potential Theory and Its Probabilistic Counterpart (Springer-Verlag, 1984)CrossRefGoogle Scholar
[4]Dvoretzky, A., Erdös, P. and Kakutani, S.. Double points of paths of Brownian motion in n-space. Acta Sci. Math. (Szeged) 12 (1950), 75–81.Google Scholar
[5]Dvoretzky, A., Erdös, P., Kakutani, S. and Taylor, S. J.. Triple points of Brownian motion in 3-space. Proc. Cambridge Philos. Soc. 53 (1957), 856–862.CrossRefGoogle Scholar
[6]Port, S. C. and Stone, C. J.. Brownian Motion and Classical Potential Theory (Academic Press, 1978).Google Scholar
[7]Shiga, T. and Watanabe, S.. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. verw. Gebiet 27 (1973), 37–46.CrossRefGoogle Scholar