  We carry out an analytical and numerical study of the motion of an isolated
vortex in thermal equilibrium, the vortex being defined as the point
singularity of a complex scalar field $\psi(\r,t)$ obeying a nonlinear
stochastic Schr\"odinger equation. Because hydrodynamic fluctuations are
included in this description, the dynamical picture of the vortex emerges as
that of both a massive particle in contact with a heat bath, and as a passive
scalar advected to a background random flow. We show that the vortex does not
execute a simple random walk and that the probability distribution of vortex
flights has non-Gaussian (exponential) tails.
