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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Discrete harmonic functions Orthants Martin Boundary

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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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  • A. Ancona: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien. phAnn. Inst. Fourier 37, (2001), no. 3, 313–338.
  • Alili, L.; Doney, R. A. Martin boundaries associated with a killed random walk. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 3, 313–338.
  • Bauman, Patricia. Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22 (1984), no. 2, 153–173.
  • Doney, R. A. The Martin boundary and ratio limit theorems for killed random walks. J. London Math. Soc. (2) 58 (1998), no. 3, 761–768.
  • Doob, J. L. Discrete potential theory and boundaries. J. Math. Mech. 8 1959 433–458; erratum 993.
  • Fayolle, Guy; Raschel, Kilian. About a possible analytic approach for walks in the quarter plane with arbitrary big jumps. C. R. Math. Acad. Sci. Paris 353 (2015), no. 2, 89–94.
  • Hunt, Richard A.; Wheeden, Richard L. Positive harmonic functions on Lipschitz domains. Trans. Amer. Math. Soc. 147 1970 507–527.
  • Ignatiouk-Robert, Irina. Martin boundary of a reflected random walk on a half-space. Probab. Theory Related Fields 148 (2010), no. 1-2, 197–245.
  • Ignatiouk-Robert, Irina; Loree, Christophe. Martin boundary of a killed random walk on a quadrant. Ann. Probab. 38 (2010), no. 3, 1106–1142.
  • Jerison, David S.; Kenig, Carlos E. Boundary value problems on Lipschitz domains. Studies in partial differential equations, 1–68, MAA Stud. Math., 23, Math. Assoc. America, Washington, DC, 1982.
  • Kuo, Hung-Ju; Trudinger, Neil S. Positive difference operators on general meshes. Duke Math. J. 83 (1996), no. 2, 415–433.
  • Kuo, Hung-Ju; Trudinger, Neil S. Evolving monotone difference operators on general space-time meshes. Duke Math. J. 91 (1998), no. 3, 587–607.
  • Kurkova, I. A.; Malyshev, V. A. Martin boundary and elliptic curves. Markov Process. Related Fields 4 (1998), no. 2, 203–272.
  • Kurkova, Irina; Raschel, Kilian. Random walks in $(\Bbb Z_ +)^ 2$ with non-zero drift absorbed at the axes. Bull. Soc. Math. France 139 (2011), no. 3, 341–387.
  • Lawler, Gregory F. Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments. Proc. London Math. Soc. (3) 63 (1991), no. 3, 552–568.
  • Mustapha, Sami. Gaussian estimates for spatially inhomogeneous random walks on ${\bf Z}^ d$. Ann. Probab. 34 (2006), no. 1, 264–283.
  • Mustapha, Sami. Gambler's ruin estimates for random walks with symmetric spatially inhomogeneous increments. Bernoulli 13 (2007), no. 1, 131–147.
  • Picardello, Massimo A.; Woess, Wolfgang. Martin boundaries of Cartesian products of Markov chains. Nagoya Math. J. 128 (1992), 153–169.
  • Raschel, Kilian. Green functions and Martin compactification for killed random walks related to $\rm SU(3)$. Electron. Commun. Probab. 15 (2010), 176–190.
  • Raschel, Kilian. Random walks in the quarter plane, discrete harmonic functions and conformal mappings. With an appendix by Sandro Franceschi. Stochastic Process. Appl. 124 (2014), no. 10, 3147–3178.
  • Safonov, Mikhail V.; Yuan, Yu. Doubling properties for second order parabolic equations. Ann. of Math. (2) 150 (1999), no. 1, 313–327.
  • Woess, Wolfgang. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3