# Long time behavior of diffusions with Markov switching

## TL;DR

This paper analyzes the long-term behavior of Ornstein-Uhlenbeck diffusions influenced by Markov switching, providing quantitative estimates and classifying the tail behavior of their stationary distributions.

## Contribution

It offers new quantitative estimates for the asymptotic behavior and a trichotomy classification of the stationary distribution tails for Markov-switching diffusions.

## Key findings

- Established quantitative estimates for long time behavior.
- Classified stationary distribution tails into heavy, exponential-like, and Gaussian-like.
- Characterized critical moments based on model parameters.

## Abstract

Let $Y$ be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite state Markov process $X$: $dY_t=-\lambda(X_t)Y_tdt+\sigma(X_t)dB_t$, $Y_0$ given. Under ergodicity condition, we get quantitative estimates for the long time behavior of $Y$. We also establish a trichotomy for the tail of the stationary distribution of $Y$: it can be heavy (only some moments are finite), exponential-like (only some exponential moments are finite) or Gaussian-like (its Laplace transform is bounded below and above by Gaussian ones). The critical moments are characterized by the parameters of the model.