  We study the Langevin dynamics for the family of spherical $p$-spin
disordered mean-field models and prove that in the limit of system size $N$
approaching infinity, the empirical state correlation and integrated response
functions converge almost surely and uniformly in time, to the non-random
unique strong solution of a pair of explicit non-linear integro-differential
equations introduced by Cugliandolo and Kurchan.
