7A Diffusion in domains with switching boundaries

In the analysis of narrow escape problems in Sect. 7.2, we assumed that the holes in the boundary of the domain were always open. However, if we view each pore as a stochastically gated ion channel, for example, then one has the problem of analyzing the diffusion equation in a domain with a randomly switching boundary (or partial boundary). Such a scenario also arises in the case of membrane diffusion within corrals, when one no longer assumes that the corral is well-mixed (Sect. 3.6). Mathematically speaking, we can view diffusion in a randomly switching environment as an example of a stochastic hybrid system involving a piecewise deterministic PDE4. We will exploit this interpretation in order to develop a method for analyzing the 1D diffusion equation with switching boundary conditions, based on discretizing space and constructing the Chapman-Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system [5]. We show how the CK equation can be used to determine the moments of the stochastic concentration in the continuum limit. We thus obtain a hierarchy of equations for the rth moments, which take the form of r-dimensional parabolic PDEs that couple to lower order moments at the boundaries. The rth moments of the stochastic PDE determine the statistics of r non-interacting Brownian particles; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment.