Dirichlet forms for singular diffusion on graphs

TL;DR

This paper introduces operators that govern singular diffusion processes on finite graphs with mass-carrying vertices, encompassing quantum graphs and discrete Laplace operators, using quadratic forms on Hilbert spaces.

Contribution

It develops a unified framework for singular diffusion on graphs via quadratic forms, extending to quantum graphs and discrete Laplace operators.

Findings

Operators for singular diffusion are characterized on finite graphs.

The framework includes quantum graphs and discrete Laplace operators.

Mathematical tools involve quadratic forms on Hilbert spaces.

Abstract

We describe operators driving the time evolution of singular diffusion on finite graphs whose vertices are allowed to carry masses. The operators are defined by the method of quadratic forms on suitable Hilbert spaces. The model also covers quantum graphs and discrete Laplace operators.