Electronic Communications in Probability

Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

Mustapha Sami, Aymen Bouaziz, and Mohamed Sifi

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Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 52, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320979

Digital Object Identifier
doi:10.1214/ECP.v20-4249

Mathematical Reviews number (MathSciNet)
MR3374302

Zentralblatt MATH identifier
1332.60067

Subjects
Primary: 60G50: Sums of independent random variables; random walks 31C35: Martin boundary theory [See also 60J50]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]

Keywords
Discrete harmonic functions Orthants Martin Boundary

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Sami, Mustapha; Bouaziz, Aymen; Sifi, Mohamed. Discrete harmonic functions on an orthant in $\mathbb{Z}^d$. Electron. Commun. Probab. 20 (2015), paper no. 52, 13 pp. doi:10.1214/ECP.v20-4249. https://projecteuclid.org/euclid.ecp/1465320979


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