Discrete harmonic functions for non-symmetric Laplace operators in the quarter plane

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of non-symmetric random walks. By solving a boundary value proble...

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Bibliographic Details
Published in:arXiv.org 2022-10
Main Author: Hoang, Viet Hung
Format: Article
Language:English
Subjects:
Boundary value problems
Harmonic functions
Laplace transforms
Operators (mathematics)
Power series
Random walk
Transition probabilities
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Summary:We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of non-symmetric random walks. By solving a boundary value problem for generating functions of harmonic functions, we deduce explicit expressions for the generating functions in terms of conformal mappings. These mappings are yielded from a conformal welding problem with quasisymmetric shift and contain information about the growth of harmonic functions. Further, we describe the set of harmonic functions as a vector space isomorphic to the space of formal power series.
ISSN:2331-8422