  Diffusion through semipermeable structures arises in a wide range of
processes in the physical and life sciences. Examples at the microscopic level
range from artificial membranes for reverse osmosis to lipid bilayers
regulating molecular transport in biological cells to chemical and electrical
gap junctions. There are also macroscopic analogs such as animal migration in
heterogeneous landscapes. It has recently been shown that one-dimensional
diffusion through a barrier with constant permeability $\kappa_0$ is equivalent
to snapping out Brownian motion (BM). The latter sews together successive
rounds of partially reflecting BMs that are restricted to either the left or
right of the barrier. Each round is killed when its Brownian local time exceeds
an exponential random variable parameterized by $\kappa_0$. A new round is then
immediately started in either direction with equal probability. In this paper
we use a combination of renewal theory, Laplace transforms and Green's function
methods to show how an extended version of snapping out BM provides a general
probabilistic framework for modeling diffusion through a semipermeable barrier.
This includes modifications of the diffusion process away from the barrier (eg.
stochastic resetting) and non-Markovian models of membrane absorption that kill
each round of partially reflected BM. The latter leads to time-dependent
permeabilities.
