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Zbl 1226.60113
Appuhamillage, Thilanka; Bokil, Vrushali; Thomann, Enrique; Waymire, Edward; Wood, Brian
Occupation and local times for skew Brownian motion with applications to dispersion across an interface.
(English)
[J] Ann. Appl. Probab. 21, No. 1, 183-214 (2011). ISSN 1050-5164

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 {\it B. Berkowitz, A. Cortis, I. Dror} and {\it 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 $D^{-}< D^+$.
[Bo Markussen (Kopenhagen)]
MSC 2000:
*60J70 Appl. of diffusion theory
60K40 Physical appl. of random processes
60G44 Martingales with continuous parameter
35C15 Integral representations of solutions of PDE
35R05 PDE with discontinuous coefficients or data
47D07 Markov semigroups of linear operators

Keywords: skew Brownian motion; advection-diffusion; local time; occupation time; elastic skew Brownian motion; stochastic order; first passage time; Fickian flux laws

Cited in: Zbl 1238.60117

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