  A stochastic theory is presented for a quantum vortex that is expected to
occur in superfluids coated on two dimensional sphere $ {\rm S}^2 $. The
starting point is the canonical equation of motion (the Kirchhoff equation) for
a point vortex, which is derived using the time-dependent Landau-Ginzburg
theory. The vortex equation, which is equivalent to the spin equation, turns
out to be the Langevin equation, from which the Fokker-Planck equation is
obtained by using the functional integral technique. The Fokker-Planck equation
is solved for several typical cases of the vortex motion by noting the specific
form of pinning potential. An extension to the non-spherical vortices is
briefly discussed for the case of the vortex on plane and pseudo-sphere.
