  We derive the thermodynamic limit of the empirical correlation and response
functions in the Langevin dynamics for spherical mixed $p$-spin disordered
mean-field models, starting uniformly within one of the spherical bands on
which the Gibbs measure concentrates at low temperature for the pure $p$-spin
models and mixed perturbations of them. We further relate the large time
asymptotics of the resulting coupled non-linear integro-differential equations,
to the geometric structure of the Gibbs measures (at low temperature), and
derive their FDT solution (at high temperature).
