  We study the effects of finite temperature on the dynamics of non-planar
vortices in the classical, two-dimensional anisotropic Heisenberg model with
XY- or easy-plane symmetry. To this end, we analyze a generalized
Landau-Lifshitz equation including additive white noise and Gilbert damping.
Using a collective variable theory with no adjustable parameters we derive an
equation of motion for the vortices with stochastic forces which are shown to
represent white noise with an effective diffusion constant linearly dependent
on temperature. We solve these stochastic equations of motion by means of a
Green's function formalism and obtain the mean vortex trajectory and its
variance. We find a non-standard time dependence for the variance of the
components perpendicular to the driving force. We compare the analytical
results with Langevin dynamics simulations and find a good agreement up to
temperatures of the order of 25% of the Kosterlitz-Thouless transition
temperature. Finally, we discuss the reasons why our approach is not
appropriate for higher temperatures as well as the discreteness effects
observed in the numerical simulations.
