Snapping out Brownian motion and interfacial diffusion

Diffusion through semipermeable interfaces has a wide range of applications, including molecular transport through biological membranes, reverse osmosis, porous media, and drug delivery. In this talk we present a probabilistic model of interfacial diffusion based on snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either side (U or V) of a semipermeable interface S. Each round is killed at the interface when its Brownian local time exceeds a random threshold. A new round is then immediately started in U with probability p or V with probability 1-p. We show that the probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the interface. In the case of an exponentially distributed local time threshold with rate k, the solution of the renewal equation satisfies the classical diffusion equation for a semi-permeable membrane with constant permeability k/2. On the other hand, if the threshold distribution is non-exponential, then the resulting permeability is a time-dependent function that tends to be heavy-tailed. We show how to incorporate stochastic resetting into the renewal equation and apply the theory to a model of synaptic receptor trafficking. 

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