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On the discrete heat equation taking values on a tree

Authors: Carl Mueller and Kijung Lee
Journal: Proc. Amer. Math. Soc. 137 (2009), 1467-1478
MSC (2000): Primary 60H15; Secondary 35R60, 35K05
Published electronically: November 18, 2008
MathSciNet review: 2465673
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Abstract: This paper was motivated by the question of studying PDE or stochastic PDE taking values on nonsmooth spaces. This is a hard problem in general, so we concentrate on a test case: the heat equation taking values on the union of rays emanating from the origin. We construct a series of discrete approximation to the solution and show that they converge to a limit. Unfortunately, we do not know if the limit is uniqueness. Our tools are probabilistic, exploiting the well-known connection between Brownian motion and the heat equation.

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Additional Information

Carl Mueller
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Kijung Lee
Affiliation: Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea

Keywords: Heat equation, white noise, stochastic partial differential equations
Received by editor(s): January 9, 2008
Published electronically: November 18, 2008
Additional Notes: The first author was supported by NSF and NSA grants
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.