  A large class of linear memory differential equations in one dimension, where
the evolution depends on the whole history, can be equivalently described as a
projection of a Markov process living in a higher dimensional space. Starting
with such a memory equation, we propose an explicit construction of the
corresponding Markov process. From a physical point of view the Markov process
can be understood as a change of the type of some quasiparticles along one-way
loops. Typically, the arising Markov process does not have the detailed balance
property. The method leads to a more realistic modeling of memory equations.
Moreover, it carries over the large number of investigation tools for Markov
processes to memory equations, like the calculation of the equilibrium state,
the asymptotic behavior and so on. The method can be used for an approximative
solution of some degenerate memory equations like delay differential equations.
