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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On minimal parabolic functions
and time-homogeneous parabolic $h$-transforms


Authors: Krzysztof Burdzy and Thomas S. Salisbury
Journal: Trans. Amer. Math. Soc. 351 (1999), 3499-3531
MSC (1991): Primary 31C35, 60J50; Secondary 31B05, 60J45, 60J65
Published electronically: March 29, 1999
MathSciNet review: 1661309
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Abstract | References | Similar Articles | Additional Information

Abstract: Does a minimal harmonic function $h$ remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes $D\subset \mathbb{R}^{d}$ of variable width and minimal harmonic functions $h$ corresponding to the boundary point of $D$ ``at infinity.'' Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and $f$ is a Lipschitz function. The answer to the question is affirmative if and only if $\int ^{\infty }f^{3}(u)du = \infty $. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous.


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Additional Information

Krzysztof Burdzy
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350
Email: burdzy@math.washington.edu

Thomas S. Salisbury
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Email: salt@nexus.yorku.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02471-X
PII: S 0002-9947(99)02471-X
Keywords: Martin boundary, harmonic functions, parabolic functions, Brownian motion, $h$-transforms
Received by editor(s): December 9, 1997
Received by editor(s) in revised form: November 6, 1998
Published electronically: March 29, 1999
Additional Notes: The first author was supported in part by NSF grant DMS-9700721.
The second author was supported in part by a grant from NSERC. A portion of this research took place during his stay at the Fields Institute.
Article copyright: © Copyright 1999 American Mathematical Society