  The theory of ``Markov-up'' processes is being developed. This is a new class
of stochastic processes with ``partial'' markovian features; it could also be
called ``one-sided Markov''. Such a behavior may be found in the real world and
in technical literature. So, naturally, phenomena of this type require special
mathematical models. This paper follows the proposal of such a model published
recently where the issue of recurrence and positive recurrence was
investigated. In this article the exponential recurrence of the same process is
established under appropriate assumptions, which potentially leads to an
exponential rate of convergence to the invariant measure.
