Foundations of Potential Theory on Finite Non-Symmetric Networks

In this article, we develop a rigorous function theory for finite sets based on the properties of a non-symmetric Laplacian matrix Δ. We consider a finite connected network {X,t(x,y)} where the transition functions t(x,y) are non-negative and satisfy t(x,y)=0 if, and only if, t(y,x)=0. We do not assume the symmetry of the transition functions, thereby allowing for a broader application to directed networks and non-reversible Markov chains. Unlike symmetric networks, the non-symmetric case requires a careful treatment of the transition function t(x,y) using the Perron–Frobenius Theorem to establish existence and uniqueness results. We define and characterize classes of subharmonic and harmonic functions, establishing discrete analogs of fundamental principles, including the Minimum Principle and the Maximum Principle. A central result of the paper is the solvability of the general Dirichlet-Poisson problem, whose proof is based on showing that the Laplacian Δ of a connected network consisting of n vertices has rank n−1. This theoretical framework provides the necessary mathematical basis for solving boundary value problems and understanding the global behavior of functions on finite networks without the assumption of symmetry.