Occupation and local times for skew Brownian motion with applications to dispersion across an interface

Abstract

Advective skew dispersion is a natural Markov process defined by a diffusion with drift across an interface of jump discontinuity in a piecewise constant diffusion coefficient. In the absence of drift, this process may be represented as a function of α-skew Brownian motion for a uniquely determined value of α=α^∗; see Ramirez et al. [Multiscale Model. Simul. 5 (2006) 786–801]. In the present paper, the analysis is extended to the case of nonzero drift. A determination of the (joint) distributions of key functionals of standard skew Brownian motion together with some associated probabilistic semigroup and local time theory is given for these purposes. An application to the dispersion of a solute concentration across an interface is provided that explains certain symmetries and asymmetries in recently reported laboratory experiments conducted at Lawrence–Livermore Berkeley Labs by Berkowitz et al. [Water Resour. Res. 45 (2009) W02201].