  For stochastic systems with discrete time delay, the Fokker-Planck equation
(FPE) of the one-time probability density function (PDF) does not provide a
complete, self-contained probabilistic description. It explicitly involves the
two-time PDF, and represents, in fact, only the first member of an infinite
hierarchy. We here introduce a new approach to obtain a Fokker-Planck
description by using a Markovian embedding technique and a subsequent limiting
procedure. On this way, we find a closed, complete FPE in an
infinite-dimensional space, from which one can derive a hierarchy of FPEs.
While the first member is the well-known FPE for the one-time PDF, we obtain,
as second member, a new representation of the equation for the two-time PDF.
From a conceptual point of view, our approach is simpler than earlier
derivations and it yields interesting insight into both, the physical meaning,
and the mathematical structure of delayed processes. We further propose an
approximation for the two-time PDF, which is a central quantity in the
description of these non-Markovian systems as it directly gives the correlation
between the present and the delayed state. Application to a prototypical
bistable system reveals that this approximation captures the non-trivial
effects induced by the delay remarkably well, despite its surprisingly simple
form. Moreover, it outperforms earlier approaches for the one-time PDF in the
regime of large delays.
