Аннотация:A simple model of the new notion of ``Markov up'' processes is proposed; its positive recurrence and ergodic properties are shown under the appropriate conditions. A one-dimensional process in discrete time moves upwards as if it were Markov, and goes down in a more complicated way, remembering all its past from the moment of its ``u-turn'' down. Also, it is assumed that in some sense its move downwards becomes more and more likely after each step in this direction.