On the Coupling Property and the Liouville Theorem for Ornstein-Uhlenbeck Processes

TL;DR

This paper establishes coupling properties and Liouville theorems for Ornstein-Uhlenbeck processes driven by compound Poisson processes, providing explicit coupling methods and gradient estimates based on the process's characteristic exponent.

Contribution

It introduces explicit coupling techniques and gradient estimates for Ornstein-Uhlenbeck processes using their characteristic exponents, extending understanding of their probabilistic behavior.

Findings

Successful coupling for Ornstein-Uhlenbeck processes established

Explicit coupling property derived from the characteristic exponent

Gradient estimates obtained via the symbol of the process

Abstract

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol.