Explicit solutions are presented in the Laplace and time domains for a one-variable Fokker-Planck equation governing the probability density of a random walker moving in a confining potential. Illustrative applications are discussed in two unrelated physical contexts: quantum yields in a doped molecular crystal or photosynthetic system, and the motion of signal receptor clusters on the surface of a cell encountered in a problem in immunology. An interesting counterintuitive effect concerning the consequences of confinement is found in the former, and some insights into the driving force for microcluster centralization are gathered in the latter application.