Laplace and bi-Laplace equations for directed networks and Markov chains

The networks of this – primarily (but not exclusively) expository compendium are strongly connected, finite directed graphs X, where each oriented edge (x,y) is equipped with a positive weight (conductance) a(x,y). We assuming symmetry function, and in general we do require that along (x,y), also (y,x) an edge. weights give rise to difference operator, the normalised version which consider as our Laplace operator. It associated Markov chain state space X. A non-empty subset X designated boundary. provide systematic exposition different types equations, starting Poisson equation, Dirichlet problem Neumann problem. For latter, discuss definition outer normal derivatives. then pass equations involving potentials, thereby addressing Robin boundary Next, study bi-Laplacian equations: iterated bi-Laplace problems, ”plate equation”. turns out map non-trivial interest. concludes two detailed examples.