On exponential recurrence of "Markov-up'' processes

TL;DR

This paper develops the theory of 'Markov-up' processes, a new class of stochastic processes with one-sided Markovian features, establishing their exponential recurrence and potential for rapid convergence to equilibrium.

Contribution

It introduces the concept of 'Markov-up' processes and proves their exponential recurrence under certain conditions, advancing the mathematical understanding of these processes.

Findings

Proves exponential recurrence of 'Markov-up' processes.

Establishes conditions for exponential rate of convergence.

Provides a foundation for modeling phenomena with one-sided Markovian behavior.

Abstract

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.