## Electronic Communications in Probability

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

#### 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

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

Rights

#### 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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