Open Access
2015 Discrete harmonic functions on an orthant in $\mathbb{Z}^d$
Mustapha Sami, Aymen Bouaziz, Mohamed Sifi
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Electron. Commun. Probab. 20: 1-13 (2015). DOI: 10.1214/ECP.v20-4249


We give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive discrete harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed at the boundary of an orthant in Zd. Our methodsallow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.


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Mustapha Sami. Aymen Bouaziz. Mohamed Sifi. "Discrete harmonic functions on an orthant in $\mathbb{Z}^d$." Electron. Commun. Probab. 20 1 - 13, 2015.


Accepted: 18 July 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1332.60067
MathSciNet: MR3374302
Digital Object Identifier: 10.1214/ECP.v20-4249

Primary: 31C35 , 60G50
Secondary: 30F10 , 60G40

Keywords: Discrete harmonic functions , Martin boundary , Orthants

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