On polynomial recurrence property of "Markov-up" processes

TL;DR

This paper investigates polynomial recurrence properties of Markov-up processes on non-negative integers, extending previous work on exponential recurrence by incorporating memory effects and new assumptions.

Contribution

It introduces polynomial recurrence analysis for Markov-up processes with memory, expanding understanding beyond classical Markov processes.

Findings

Established polynomial recurrence conditions under new assumptions

Extended previous exponential recurrence results to polynomial cases

Analyzed the impact of memory on recurrence properties

Abstract

This work is a continuation of [Kalikaeva, MPRF, 23(2):225-240]. The object of study is ``Markov-up processes'' on $\mathbb Z_+$ and the moment of downcrossing a certain barrier. The processes considered in this paper differ from Markov ones by the presence of a memory in certain parts of the trajectory. In our previous paper [Kalikaeva, MPRF, 23(2):225-240] exponential recurrence conditions were established. In this paper polynomial recurrence properties are considered under certain new assumptions.