  We analyze the mean time t_{app} that a randomly moving particle spends in a
bounded domain (sphere) before it escapes through a small window in the
domain's boundary. A particle is assumed to diffuse freely in the bulk until it
approaches the surface of the domain where it becomes weakly adsorbed, and then
wanders diffusively along the boundary for a random time until it desorbs back
to the bulk, and etc. Using a mean-field approximation, we define t_{app}
analytically as a function of the bulk and surface diffusion coefficients, the
mean time it spends in the bulk between two consecutive arrivals to the surface
and the mean time it wanders on the surface within a single round of the
surface diffusion.

Soft Condensed Matter;Biological Physics;Chemical Physics