Advection–Dispersion Across Interfaces

This article concerns a systemic manifestation of small scale in- terfacial 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 gov- erning the conservative, advective-dispersive transport of mass concentra- tions in divergence form, the specific interfacial heterogeneities are intro- duced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic for- mulation 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 ap- plications involving one-dimensional models having piecewise constant co- efficients 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.