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Discrete harmonic functions in Lipschitz domains. (English) Zbl 1422.60076
Summary: We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in \(\mathbb{Z} ^d\). Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.
MSC:
60G50 Sums of independent random variables; random walks
31C35 Martin boundary theory
60G40 Stopping times; optimal stopping problems; gambling theory
30F10 Compact Riemann surfaces and uniformization
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