Sticky diffusions on graphs

TL;DR

This paper studies diffusion processes on metric graphs with sticky membranes at vertices, analyzing their behavior as diffusion speed increases and permeability decreases, and establishes the governing semigroup.

Contribution

It introduces a model of sticky diffusions on graphs and characterizes its asymptotic behavior under extreme parameter limits.

Findings

The process is governed by a Feller semigroup.

As diffusion speed increases and permeability decreases, the process exhibits specific asymptotic behavior.

The paper provides mathematical proofs for the semigroup and asymptotics.

Abstract

We consider diffusion processes on metric graphs with semipermeable sticky membranes in each vertex. We prove that the process is governed by a Feller semigroup and find its asymptotic behavior as diffusion's speed increases to infinity with the same rate as permeability coefficients decreases to zero.