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

Abstract

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. Contents 1. Introduction 1 2. Study of the kernel 8 3. Conformal welding 18 4. Proof of the main theorems 26 5. Small jump random walks 29 References 33