# Bi-continuous semigroups for flows on infinite networks

## TL;DR

This paper investigates transport processes on infinite metric graphs with variable velocities and complex boundary conditions, employing bi-continuous semigroup theory to establish well-posedness in an $L^{\,\infty}$-framework.

## Contribution

It introduces a novel application of bi-continuous semigroup theory to analyze transport on infinite networks with stochastic boundary conditions.

## Key findings

- Established well-posedness of transport equations on infinite graphs
- Extended semigroup theory to handle non-constant velocities and stochastic matrices
- Provided conditions for the existence and uniqueness of solutions

## Abstract

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $\\mathrm{L}^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.