Аннотация:This paper continues developing the theory of ``Markov-up'' processes. This is a new class of stochastic processes with ``partial'' markovian features; it could also be called ``one-sided Markov''. Naturally, phenomena of this type require special mathematical models. In this article exponential recurrence of the same process is established under appropriate assumptions, which potentially leads to an exponential rate of convergence to the invariant measure.