  For classical Brownian systems driven out of equilibrium we derive
inhomogeneous two-time correlation functions from functional differentiation of
the one-body density and current with respect to external fields. In order to
allow for appropriate freedom upon building the derivatives, we formally
supplement the Smoluchowski dynamics by a source term, which vanishes at the
physical solution. These techniques are applied to obtain a complete set of
dynamic Ornstein-Zernike equations, which serve for the development of
approximation schemes. The rules of functional calculus lead naturally to
non-Markovian equations of motion for the two-time correlators. Memory
functions are identified as functional derivatives of a unique space- and
time-nonlocal dissipation power functional.
