Constructing discrete harmonic functions in wedges

Abstract

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

arXiv comment42 pages, 8 figures

References in corpus (14)

• Walks with small steps in the quarter plane
• Random walks in cones
• Littelmann paths and brownian paths
• On the functions counting walks with small steps in the quarter plane
• On the nature of the generating series of walks in the quarter plane
• Martin boundary of a killed random walk on a quadrant
• Random walks in the quarter plane, discrete harmonic functions and conformal mappings
• Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group
• Weighted Lattice Walks and Universality Classes
• About a possible analytic approach for walks in the quarter plane with arbitrary big jumps
• Gaussian estimates for spatially inhomogeneous random walks on ${\mathbf{Z}}^d$
• $t$-Martin boundary of killed random walks in the quadrant
• Plane bipolar orientations and quadrant walks
• Polyharmonic functions and random processes in cones