Splitting probabilities and interfacial territory covered by two-dimensional and three-dimensional surface-mediated diffusion

We consider the mean territory covered by a particle that performs surface-mediated diffusion inside a spherical confining domain (in two and three dimensions) before exit through an opening on the surface. This quantity can be expressed in terms of the splitting probability between two targets on the surface. We derive a general formula that relates this splitting probability to the mean first passage time to a single target that has been recently calculated for such a surface-mediated diffusion process. This formula is exact for pointlike targets and is shown to be accurate for extended targets. The mean covered territory is then found and analyzed for an arbitrary extension of the exit region in both two- and three-dimensional spherical domains.