Dirichlet problem on locally finite graphs (revised and extended version)

Abstract

We study existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let $B$ be a subset of the vertices of a connected graph $G$. The Dirichlet problem asks for a function whose discrete Laplacian is prescribed on $V\setminus B$ and whose values are prescribed on $B$. Each infinite connected component of $G\setminus B$ is called an end of $G$ relative to $B$. On finite graphs we show that the solution is the limit of an averaging process, with an explicit geometric rate given by the spectrum of the random walk, and is also the limit of the heat flow and the minimizer of an energy functional. On infinite graphs we prove two existence theorems: one with no assumption on vertex degrees, under a finite-width condition and summable data, and one for bounded data on graphs of bounded degree, where the solution is also unique among bounded functions. If there are no ends, the Dirichlet problem has exactly one solution. If there are $l$ ends, the solution set, when nonempty, has dimension at least $l$; each end carries a positive harmonic function vanishing on $B$. Calling an end parabolic when the random walk started in it hits $B$ almost surely, we show that this function can be taken bounded exactly on non-parabolic ends and is necessarily unbounded on parabolic ones, a discrete analogue of a theorem of Li and Tam. Consequently, bounded solutions are unique if and only if every end is parabolic.