In this paper, the author investigates the scaling limit of a partial difference equation on the d dimensional integer lattice Z^d, corresponding to a translation invariant random walk perturbed by a random vector field. In the case when the translation invariant walk scales to a Cauchy process he proves convergence to an effective equation on R^d. The effective equation corresponds to a Cauchy process perturbed by a constant vector field. In the case when the translation invariant walk scales to Brownian motion he shows that the scaling limit, if it exists, depends on dimension. For d=1,2 he provides evidence that the scaling limit cannot be diffusion.