Discrete harmonic functions on an orthant in \(\mathbb{Z}^d\). (English) Zbl 1332.60067

Sami, Mustapha; Bouaziz, Aymen; Sifi, Mohamed

Electron. Commun. Probab. 20, Paper No. 52, 13 p. (2015).

Summary: 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 \(\mathbb Z^d\). Our methods allow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.

MSC:

60G50	Sums of independent random variables; random walks
60J50	Boundary theory for Markov processes
31C35	Martin boundary theory
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

random walk; discrete harmonic functions; orthants; Martin boundary