We consider the time correlation function of observables pertaining to a (quantum subsystem +bath), where the bath is coupled to a reservoir with many degrees of freedom. Integrating over the coordinates of this reservoir and assuming no initial correlations between the (quantum subsystem+bath) and the reservoir, we obtain an expression for the time correlation function that contains an influence functional. We then take the semiclassical and Fokker-Planck limits while modeling the reservoir with an Ohmic continuum of harmonic oscillators coupled bilinearily to the coordinates of the bath. The semiclassical limit is taken using a variant of Pechukas＇ stationary phase analysis of the reduced propagator that yields a time correlation function written in terms of connected "classical" paths. These paths are got by solving the concatenation of several short-time interval Pechukas equations; as a result, the determination of these paths is more feasible than the determination of the "classical" path associated with a single long-time interval Pechukas equation. This concatenation includes the dissipative and stochastic forces associated with a classical Brownian particle. We then use decoherence arguments derived from an inspection of the influence functional to eliminate the phase interference structure of the bath. This elimination yields a mixed quantum-classical time correlation function that can be evaluated using nonadiabatic mixed quantum-classical dynamics schemes similar to those proposed recently by Webster and Tully.