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

DOI:
https://doi.org/10.1090/S0002-9939-08-09748-7

Published electronically:
November 18, 2008

MathSciNet review:
2465673

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Abstract | References | Similar Articles | Additional Information

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

Email:
cmlr@math.rochester.edu

**Kijung Lee**

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

Email:
kijung@yonsei.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-08-09748-7

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.