Random walks in two-dimensional random environments with constrained drift forces

Random walks in two-dimensional environments with a positionally random drift force are analyzed. If the force is constrained to be divergence-free, then the mean-square displacement is superdiffusive 〈${x}^{{2}^{\mathrm{}(t)}}$\ifmmode\bar\else\textasciimacron\fi{}〉\ensuremath{\sim}t (lnt${)}^{1/2}$. If in addition the force has a component which is curl-free, there are two cases: If the two components are independent, the long-time behavior is diffusive with only logarithmic corrections; on the other hand, if the two components of the force are, respectively, parallel and perpendicular to the gradients of a single strongly fluctuating potential, the long-time behavior is subdiffusive and dominated by the longitudinal part.