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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Finely harmonic functions with finite Dirichlet integral with respect to the Green measure


Author: Bernt Øksendal
Journal: Trans. Amer. Math. Soc. 287 (1985), 687-700
MSC: Primary 60J45; Secondary 31C05, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1985-0768734-3
MathSciNet review: 768734
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Abstract: We consider finely harmonic functions $ h$ on a fine, Greenian domain $ V \subset {{\mathbf{R}}^d}$ with finite Dirichlet integral wrt $ Gm$, i.e. $ (\ast)$

$\displaystyle \int_V\vert\nabla h(y)\vert^2G(x,y)\,dm(y) < \infty \quad {\text{for}}\;x \in V,$

where $ m$ denotes the Lebesgue measure, $ G(x,y)$ the Green function. We use Brownian motion and stochastic calculus to prove that such functions $ h$ always have boundary values $ {h^\ast}$ along a.a. Brownian paths. This partially extends results by Doob, Brelot and Godefroid, who considered ordinary harmonic functions with finite Dirichlet integral wrt $ m$ and Green lines instead of Brownian paths.

As a consequence of Theorem 1 we obtain several properties equivalent to $ ( \ast )$, one of these being that $ h$ is the harmonic extension to $ V$ of a random "boundary" function $ {h^\ast}$ (of a certain type), i.e. $ h(x) = {E^x}[{h^\ast}]$ for all $ x \in V$. Another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying $ ( \ast )$. This is in contrast with the situation for finely harmonic functions with finite Dirichlet integral wrt $ m$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768734-3
Article copyright: © Copyright 1985 American Mathematical Society