  We study the dynamics of a directed manifold of internal dimension D in a
d-dimensional random force field. We obtain an exact solution for $d \to
\infty$ and a Hartree approximation for finite d. They yield a Flory-like
roughness exponent $\zeta$ and a non trivial anomalous diffusion exponent $\nu$
continuously dependent on the ratio $g_{T}/g_{L}$ of divergence-free ($g_{T}$)
to potential ($g_{L}$) disorder strength. For the particle (D=0) our results
agree with previous order $\epsilon^2$ RG calculations. The time-translational
invariant dynamics for $g_{T} >0$ smoothly crosses over to the previously
studied ultrametric aging solution in the potential case.
