  Dynamics of a particle diffusing in a confinement can be seen a sequence of
bulk-diffusion-mediated hops on the confinement surface. Here, we investigate
the surface hopping propagator that describes the position of the diffusing
particle after a prescribed number of encounters with that surface. This
quantity plays the central role in diffusion-influenced reactions and
determines their most common characteristics such as the propagator, the
first-passage time distribution, and the reaction rate. We derive explicit
formulas for the surface hopping propagator and related quantities for several
Euclidean domains: half-space, circular annuli, circular cylinders, and
spherical shells. These results provide the theoretical ground for studying
diffusion-mediated surface phenomena. The behavior of the surface hopping
propagator is investigated for both "immortal" and "mortal" particles.
