  Using third-order perturbation theory, we derive a relation between the mean
divergence of the peculiar velocity given density and the density itself. Our
calculations assume Gaussian initial conditions and are valid for Gaussian
filtering of the evolved density and velocity fields. The mean velocity
divergence turns out to be a third-order polynomial in the density contrast. We
test the power spectrum dependence of the coefficients of the polynomial for
scale-free and standard CDM spectra and find it rather weak. Over scales larger
than about 5 megaparsecs, the scatter in the relation is small compared to that
introduced by random errors in the observed density and velocity fields. The
relation can be useful for recovering the peculiar velocity from the associated
density field, and also for non-linear analyses of the anisotropies of
structure in redshift surveys.
