  This article concerns a systemic manifestation of small scale interfacial
heterogeneities in large scale quantities of interest to a variety of diverse
applications spanning the earth, biological and ecological sciences. Beginning
with formulations in terms of partial differential equations governing the
conservative, advective-dispersive transport of mass concentrations in
divergence form, the specific interfacial heterogeneities are introduced in
terms of (spatial) discontinuities in the diffusion coefficient across a
lower-dimensional hypersurface. A pathway to an equivalent stochastic
formulation is then developed with special attention to the interfacial effects
in various functionals such as first passage times, occupation times and local
times. That an appreciable theory is achievable within a framework of
applications involving one-dimensional models having piecewise constant
coefficients greatly facilitates our goal of a gentle introduction to some
rather dramatic mathematical consequences of interfacial effects that can be
used to predict structure and to inform modeling.
