  We consider the time-dependent statistical distributions of diffusive
processes in relaxation to a stationary state for simple, two dimensional
chaotic models based upon random walks on a line. We show that the cumulative
functions of the hydrodynamic modes of diffusion form fractal curves in the
complex plane, with a Hausdorff dimension larger than one. In the limit of
vanishing wavenumber, we derive a simple expression of the diffusion
coefficient in terms of this Hausdorff dimension and the positive Lyapunov
exponent of the chaotic model.
