An averaging principle for fast diffusions in domains separated by semi-permeable membranes

TL;DR

This paper establishes an averaging principle for fast diffusions across domains separated by semi-permeable membranes, showing convergence to a Markov chain with transition rates related to membrane permeability and domain size.

Contribution

It introduces a novel averaging principle for diffusions with semi-permeable membranes, linking boundary conditions to Markov chain limits in a singular perturbation framework.

Findings

Diffusions converge to a Markov chain as diffusion coefficients increase.

Transition rates depend on membrane permeability and domain size.

The limit involves boundary and transmission conditions in a singular perturbation setting.

Abstract

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed towards the end of the paper.