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

We prove an averaging principle which asserts convergence of diffusions on domains separated by semi-permeable membranes, when the speed of diffusion tends 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 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, including models of neurotransmitters, kinaze activity and intracellular calcium dynamics. This is a joint work with B. Kazmierczak and M. Kunze.