  Diffusion through semipermeable interfaces has a wide range of applications,
ranging from molecular transport through biological membranes to reverse
osmosis for water purification using artificial membranes. At the
single-particle level, one-dimensional diffusion through a barrier with
constant permeability $\kappa_0$ can be modeled in terms of so-called 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 (absorbed) at the barrier 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.
It has recently been shown 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 barrier. Moreover,
generalized versions of the renewal equation can be constructed that
incorporate non-Markovian, encounter-based models of absorption. In this paper
we extend the renewal theory of snapping out BM to single-particle diffusion in
bounded domains and higher spatial dimensions. We also consider an example of
an asymmetric interface in which the directional switching after each
absorption event is biased. Finally, we show how to incorporate an
encounter-based model of absorption for single-particle diffusion through a
spherically symmetric interface. We find that, even when the same non-Markovian
model of absorption applies on either side of the interface, the resulting
permeability is an asymmetric time-dependent function with memory. Moreover,
the permeability functions tend to be heavy-tailed.
