  We study the Lagrangian trajectories of statistically isotropic, homogeneous,
and stationary divergence free spatiotemporal random vector fields. We design
this advecting Eulerian velocity field such that it gets asymptotically rough
and multifractal, both in space and time, as it is demanded by the
phenomenology of turbulence at infinite Reynolds numbers. We then solve
numerically the flow equations for a differentiable version of this field. We
observe that trajectories get also rough, characterized by nearly the same
Hurst exponent as the one of our prescribed advecting field. Moreover, even
when considering the simplest situation of the advection by a fractional
Gaussian field, we evidence in the Lagrangian framework additional intermittent
corrections. The present approach involves properly defined random fields, and
asks for a rigorous treatment that would explain our numerical findings and
deepen our understanding of this long lasting problem.
