Markov processes with product-form stationary distribution

Abstract

We consider a continuous time Markov process $(X,L)$, where $X$ jumps between a finite number of states and $L$ is a piecewise linear process with state space $\mathbb{R}^d$. The process $L$ represents an "inert drift" or "reinforcement." We find sufficient and necessary conditions for the process $(X,L)$ to have a stationary distribution of the product form, such that the marginal distribution of $L$ is Gaussian. We present a number of conjectures for processes with a similar structure but with continuous state spaces.