  Stochastic dynamics of a vortex filament obeying local induced approximation
equation plus random agitation is investigated by analytical and numerical
methods. The character of a stirring force is supposed to be a white noise with
spatial correlator concentrated at large distances comparable with size of the
loop. Dependence of the spectral function $<\mathbf{s}_\kappa ^\alpha
\mathbf{s}_\kappa ^\beta >$ of the vortex line on both the one-dimensional wave
vector $\kappa$ and intensity of the external force correlator
$<\mathbf{\zeta}_\kappa ^\alpha \mathbf{\zeta}_\kappa ^\alpha>$ was studied.
Here $\mathbf{s}_\kappa ^\alpha$ is the Fourier transform of the line element
position $\mathbf{s}^\alpha (\xi, t)$. It is shown that under the influence of
an external random force a vortex ring becomes a small tangle whose mean size
depends on external force intensity. The theoretical predictions and the
numerical results are in reasonable agreement.
