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

Adam Bobrowski

Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland

E-mail Address: a.bobrowski@pollub.pl

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Bogdan Kaźmierczak

Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland

E-mail Address: bkazmier@ippt.pan.pl

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Markus Kunze

Universität Konstanz, Fachbereich Mathematik und Statistik, 78457 Konstanz, Germany

E-mail Address: markus.kunze@uni-konstanz.de

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https://doi.org/10.1142/S0218202517500130Cited by:10 (Source: Crossref)

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 toward the end of the paper.

Communicated by E. Feireisl

Keywords:

AMSC: 47A07, 47D07, 60J70, 92C45

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