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May, 1984 Probabilistic Solution of the Dirichlet Problem for Biharmonic Functions in Discrete Space
R. J. Vanderbei
Ann. Probab. 12(2): 311-324 (May, 1984). DOI: 10.1214/aop/1176993292

Abstract

Considering difference equations in discrete space instead of differential equations in Euclidean space, we investigate a probabilistic formula for the solution of the Dirichlet problem for biharmonic functions. This formula involves the expectation of a weighted sum of the pay-offs at the successive times at which the Markov chain is in the complement of the domain. To make the infinite sum converge, we use Borel's summability method. This is interpreted probabilistically by imbedding the Markov chain into a continuous time, discrete space Markov process.

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R. J. Vanderbei. "Probabilistic Solution of the Dirichlet Problem for Biharmonic Functions in Discrete Space." Ann. Probab. 12 (2) 311 - 324, May, 1984. https://doi.org/10.1214/aop/1176993292

Information

Published: May, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0543.60079
MathSciNet: MR735840
Digital Object Identifier: 10.1214/aop/1176993292

Subjects:
Primary: 60J45
Secondary: 31B30

Keywords: Biharmonic functions , Dirichlet problem , Dynkin's formula

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 2 • May, 1984
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