  We investigate the properties of the path integral over reparametrizations (=
the boundary value of the Liouville field in open string theory). Discretizing
the path integral, we apply the Metropolis-Hastings algorithm to numerical
simulations of a proper (subordinator) stochastic process and find that typical
trajectories are not Brownian but rather have discontinuities of the type of
Levy's flights. We study a fractal structure of these trajectories and show
that their Hausdorff dimension is zero. We confirm thereby the discretization
and heuristic consideration of QCD scattering amplitudes by analytical and
numerical calculations. We also perform Monte Carlo simulations of the path
integral over reparametrization in the effective-string ansatz for a circular
Wilson loop and discuss their subtleties associated with the discretization of
Douglas' functional.
