Exact and asymptotic formulas for overdamped Brownian dynamics

Exact and asymptotic formulas relating to dynamical correlations for over-damped Brownian motion are obtained. These formulas include a generalization of the f-sum rule from the theory of quantum fluids, a formula relating the static current-current correlation to the static density-density correlation, and an asymptotic formula for the small-k behaviour of the dynamical structure factor. Known exact evaluations of the dynamical density-density correlation in some special models are used to illustrate and test the formulas.