Functions on discrete spaces - A note on Dirichlet, Poisson and Neumann
problems on infinite graphs
Abstract
By a discrete space , we mean here a collection of finite or a countable
infinite number of vertices which has a graph structure provided by a
set of edges (finite or countable infinite in number). In many cases,
the varying graph structures (connectivity type problems) are themselves
very interesting and important. However, there are some important
examples where the study of the intricate role of functions on is
essential (example: potential functions, effective resistance, Kirchhoff
problem in electrical networks; and escape probability, Dirichlet
functions, hitting time in random walks). In this survey article we
review a part of the function theory developed by some researchers in
this field and present a cohesive narrative. We have placed special
emphasis on different discrete versions of the Dirichlet problem, the
Neumann problem and the Poisson equation.