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

TL;DR

This paper systematically explores Laplace and bi-Laplace equations on strongly connected directed graphs with weighted edges, linking them to Markov chains, boundary value problems, and potential theory, including novel insights into the bi-Laplace Dirichlet to Neumann map.

Contribution

It provides a comprehensive exposition of Laplace and bi-Laplace equations on directed networks, including boundary problems and the bi-Laplace Dirichlet to Neumann map, with detailed examples.

Findings

Analysis of Laplace equations on directed graphs with non-symmetric weights.

Extension to bi-Laplace equations and boundary value problems.

Insights into the bi-Laplace Dirichlet to Neumann map.

Abstract

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