On the discrete heat equation taking values on a tree
Authors:
Carl Mueller and Kijung Lee
Journal:
Proc. Amer. Math. Soc. 137 (2009), 14671478
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 wellknown connection between Brownian motion and the heat equation.
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 Kai Lai Chung.
Elementary probability theory with stochastic processes. Third edition. Undergraduate Texts in Mathematics. SpringerVerlag, New YorkHeidelberg, 1979. MR 560506 (81k:60002)
 [Cra91]
 Michael Cranston.
Gradient estimates on manifolds using coupling. J. Funct. Anal., 99(1):110124, 1991. MR 1120916 (93a:58175)
 [DE88]
 M. Doi and S.F. Edwards.
The theory of polymer dynamics, volume 73 of The International Series of Monographs in Physics. Oxford University Press, Oxford, 1988.
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 Richard Durrett.
Probability: Theory and examples. Duxbury Press, Belmont, CA, second edition, 1996. MR 1609153 (98m:60001)
 [EF01]
 J. Eells and B. Fuglede.
Harmonic maps between Riemannian polyhedra, volume 142 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001. With a preface by M. Gromov. MR 1848068 (2002h:58017)
 [Eva98]
 Lawrence C. Evans.
Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
 [Fel68]
 William Feller.
An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley & Sons Inc., New York, 1968. MR 0228020 (37:3604)
 [Fun83]
 Tadahisa Funaki.
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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 120749, Republic of Korea
Email:
kijung@yonsei.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002993908097487
PII:
S 00029939(08)097487
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.
