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

Abstract

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.

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

“…The similarity between the principle just described and the main result of the present paper is not accidental: whereas in [15,21] diffusion is fast in all directions, the thin layer approximation, as was already noted, is equivalent to an analysis of diffusion that is fast only in one direction.…”

“…The proposition that such processes are obtained by the thin layer approximation is of interest in itself, and provides a link between notions of jump intensities and permeability coefficients, i.e., shows again that transmission conditions and terms describing jumps play complementary roles (cf. Remark 2.1 in [21], and the discussion in Sect. 6 of [19]).…”

“…Coming back to conditions of type (2.3), it is worth noting that they were extensively used in the papers [15,21], devoted to the so-called averaging principle. This principle says that fast diffusions on a graph with edges separated by semi-permeable membranes situated on its vertices, or in the higher-dimensional regions separated by such membranes, are well approximated by a Markov chain with state-space constructed by identifying each edge with a point, or lumping all points of a region into one point.…”

“…These conditions describe a stochastic mechanism in which a particle in the upper layer may filter through the membrane to the lower layer, and vice versa. The functions α and β are permeability coefficients: the larger is α in a given subset of the membrane, the shorter is the average time needed for a particle to filter through this part of the membrane from the lower to the upper layer, and the larger is the β the shorter is the average time for a particle to filter from the upper to the lower layer (see [21,25,59], see also [17] p. 66 and the references given there). For our subsequent analysis (i.e., our generation and convergence results), the assumption that α and β are non-negative is not needed; but a probabilistic interpretation of transmission conditions (2.3) with negative α and β is less pleasing.…”

Lord Kelvin and Andrey Andreyevich Markov in a Queue with Single Server

“…We are using a dierent method, i.e., Lord Kelvin's method of images that has been developed as a way to deal with boundary conditions in semigroup theory in [1923] and [24]. For yet dierent ways of dealing with boundary conditions see [25] and [26].…”