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Occupation and local times for skew Brownian motion with applications to dispersion across an interface. (English) Zbl 1226.60113
Loosely speaking a skew Brownian motion is a Brownian motion with two different diffusion coefficients $$D^-$$ and $$D^+$$ on the negative and positive half-lines, respectively. The paper provides new theoretical results for functionals of skew Brownian motions and its associated semi-group theory. In particular, the trivariate density of position, the symmetric local time at zero and the occupation time on the positive half-line are obtained via a Feynman-Kac formula for an elastic skew Brownian motion. Also, an elastic change of measure is devised to obtain an integral representation of the transition density for a skew Brownian motion with non-zero drift. Besides this, several other distributional results on functionals of skew Brownian motions are given. The paper concludes with a specific application to dispersion across an interface. In an actual physical experiment by B. Berkowitz, A. Cortis, I. Dror and H. Scher [“Laboratory experiments on dispersive transport across interfaces: the role of flow direction”, Water Resour. Res. 45, W02201 (2009), doi:10.1029/2008WR007342], it was experimentally observed that fine to coarse breakthrough is faster than coarse to fine breakthrough. The theoretical results given in the present paper explain this phenomenon within the framework of Fickian flux laws. Here, fine and course media are characterized by their relative dispersion rates $$D^{-}< D^+$$.

MSC:
 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K40 Other physical applications of random processes 60G44 Martingales with continuous parameter 60J55 Local time and additive functionals 35C15 Integral representations of solutions to PDEs 35R05 PDEs with low regular coefficients and/or low regular data 47D07 Markov semigroups and applications to diffusion processes
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