Stationary distributions for jump processes with memory

TL;DR

This paper studies a jump process influenced by a memory component, proving that its stationary distribution is a product of a uniform measure on the circle and a Gaussian on the real line.

Contribution

It introduces a jump process with a memory-dependent jump measure and characterizes its stationary distribution explicitly.

Findings

Stationary distribution is a product of uniform and Gaussian measures.

The process's long-term behavior converges to this stationary distribution.

The model combines jump dynamics with memory effects on a circular and linear state space.

Abstract

We analyze a jump processes $Z$ with a jump measure 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.