# Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by L\'evy processes

## TL;DR

This paper investigates the sharp transition (cut-off phenomenon) in convergence to equilibrium for high-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes, revealing conditions for its occurrence and properties.

## Contribution

It establishes the existence of the cut-off phenomenon for these processes under total variation distance, including reversible and non-reversible cases, and analyzes related processes.

## Key findings

- Cut-off phenomenon occurs under total variation distance for these processes.
- Profile function exists in reversible cases, may exist in non-reversible cases.
- Cut-off phenomena are characterized for average and superposition processes.

## Abstract

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by L\'evy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.