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Estimates for Differences and Harnack Inequality for Difference Operators Coming From Random Walks with Symmetric, Spatially Inhomogeneous, Increments
Author(s) -
Lawler Gregory F.
Publication year - 1991
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-63.3.552
Subject(s) - harnack's inequality , mathematics , harnack's principle , subharmonic function , random walk , harmonic function , mathematical analysis , harmonic , homogeneous , pure mathematics , inequality , statistics , combinatorics , physics , quantum mechanics
Difference operators arising from random walks with symmetric increments are studied. If the random walk is spatially homogeneous, then estimates of the first and second differences of harmonic functions are given and a Harnack inequality is proved. In the spatially inhomogeneous case, a Harnack inequality for superharmonic functions is proved, giving a discrete version of a result of Krylov and Safonov. This is used to give an estimate for differences of harmonic functions and applied to show existence of harmonic measure for spatially inhomogeneous walks.