On Markov--up processes and their recurrence properties
Alexander Yu. Veretennikov, Maria Veretennikova · Warwick Research Archive Portal (University of Warwick) · 2021
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 probable after each step in this direction.