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Transactions of the American Mathematical Society

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



A Martin boundary in the plane

Author: Thomas S. Salisbury
Journal: Trans. Amer. Math. Soc. 293 (1986), 623-642
MSC: Primary 60J50; Secondary 31C35
MathSciNet review: 816315
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Abstract: Let $ E$ be an open connected subset of Euclidean space, with a Green function, and let $ \lambda $ be harmonic measure on the Martin boundary $ \Delta $ of $ E$. We will show that, except for a $ \lambda \otimes \lambda $-null set of $ (x,y) \in {\Delta ^2}$, $ x$ is an entrance point for Brownian motion conditioned to leave $ E$ at $ y$. R. S. Martin gave examples in dimension $ 3$ or higher, for which there exist minimal accessible Martin boundary points $ x \ne y$ for which this condition fails. We will give a similar example in dimension $ 2$.

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Keywords: Conditional Brownian motion, Martin boundaries
Article copyright: © Copyright 1986 American Mathematical Society