  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.
