Skew Disperson and Continuity of Local Time

Results are provided that highlight the effect of interfacial discontinuities in the diffusion coefficient on the behavior of certain basic functionals of the diffusion, such as local times and occupation times, extending previous results in (Appuhamillage et al., Ann Appl Probab 21:183–214, 2011; Water Resour Res 46:W07511, 2009) on the behavior of first passage times. The main goal is to obtain a characterization of large scale parameters and behavior by an analysis at the fine scale of stochastic particle motions. In particular, considering particle concentration modeled by a diffusion equation with piecewise constant diffusion coefficient, it is shown that the continuity of a natural modification of local time is the individual (stochastic) particle scale equivalent to continuity of flux at the scale of the (macroscopic) particle concentrations. Consequences of this involve the determination of a skewness transmission probability in the presence of an interface, as well as corollaries concerning interfacial effects on occupation time of the associated stochastic particles.