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Advection-dispersion across interfaces. (English) Zbl 1331.60003

Summary: 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.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J55 Local time and additive functionals
60J65 Brownian motion
76R50 Diffusion
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Full Text: DOI arXiv Euclid

References:

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