  In this chapter, we review our recent work on first passage time (FPT)
problems for absorption by a target whose interface is semipermeable. For
pedagogical reasons, we focus on a single Brownian particle searching for a
single target in a bounded domain. We begin by writing down the forward
diffusion equation for the target problem, and define various quantities of
interest such as the survival probability, absorption flux, and the FPT
density. We also present a general method of solution based on Green's
functions and the spectral decomposition of so-called Dirichlet-to-Neumann
(D-to-N) operators. We then use an encounter-based approach to extend the
theory to the case of non-Markovian absorption within the target interior.
Encounter-based models consider the joint probability density or generalized
propagator for particle position and the amount of particle-target contact time
prior to absorption. In the case of a partially absorbing target interior, the
contact time is given by a Brownian functional known as the occupation time.
Finally, we develop a more general probabilistic model of single-particle
diffusion through semi-permeable interfaces, by combining the encounter-based
approach with so-called snapping out Brownian motion (BM). Snapping out BM sews
together successive rounds of partially absorbing BMs that are restricted to
either the interior or the exterior of the semipermeable interface. The rule
for terminating each round is implemented using encounter-based model of
partially absorbing BM. We show that this results in a time-dependent
permeability that can be heavy-tailed.
