The Annals of Probability

Probabilistic Solution of the Dirichlet Problem for Biharmonic Functions in Discrete Space

R. J. Vanderbei

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 12, Number 2 (1984), 311-324.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993292

Digital Object Identifier
doi:10.1214/aop/1176993292

Mathematical Reviews number (MathSciNet)
MR735840

Zentralblatt MATH identifier
0543.60079

JSTOR
links.jstor.org

Subjects
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 31B30: Biharmonic and polyharmonic equations and functions

Keywords
Biharmonic functions Dirichlet problem Dynkin's formula

Citation

Vanderbei, R. J. Probabilistic Solution of the Dirichlet Problem for Biharmonic Functions in Discrete Space. Ann. Probab. 12 (1984), no. 2, 311--324. doi:10.1214/aop/1176993292. https://projecteuclid.org/euclid.aop/1176993292


Export citation