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The Local Diffusions of a Harmonic Sheaf

Published online by Cambridge University Press:  20 November 2018

Donald A. Dawson
Donald A. Dawson*
Affiliation:
McGill University
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It is well known that the Laplacian operator is the infinitesimal generator of Brownian motion in Rn. Moreover, the classical harmonic measures are the hitting measures of Brownian motion. In other words, there is a natural correspondence between the Brownian motion and the classical harmonic functions. In this paper we will show that any family of abstract harmonic functions satisfying the axioms of M. Brelot [4] are annihilated by the infinitesimal generator of a diffusion, and that the corresponding harmonic measures are the hitting measures of the diffusion. This answers a question raised by P. A. Meyer [11] who has proved the existence of a Markov semi-group associated with a harmonic sheaf.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 3 , April 1965 , pp. 307 - 316
Copyright
Copyright © Canadian Mathematical Society 1965

References

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