Stationary distributions for jump processes with inert drift

TL;DR

This paper studies a class of jump processes with inert drift, establishing their stationary distribution as a product of uniform and Gaussian measures, contributing to understanding their long-term behavior.

Contribution

It provides a rigorous analysis of the stationary distribution for jump processes with inert drift, a novel result in stochastic process theory.

Findings

Stationary distribution is the product of uniform and Gaussian measures.

The process's long-term behavior is characterized explicitly.

The analysis applies to processes on the circle and real line.

Abstract

We analyze jump processes $Z$ with ``inert drift'' determined by a ``memory'' process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.