Well-Posedness and Symmetries of Strongly Coupled Network Equations

TL;DR

This paper studies a generalized diffusion process on networks with feedback effects, analyzing well-posedness, solution properties, and symmetries, with applications in neurobiology and quantum systems.

Contribution

It extends existing models by incorporating feedback between non-adjacent edges, providing new insights into well-posedness and symmetry properties of network equations.

Findings

Established well-posedness of the generalized network diffusion model

Analyzed contractivity and positivity of solutions

Identified symmetry conditions related to invariance of subspaces

Abstract

We consider a diffusion process on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schroedinger equations on a quantum graph are discussed.