# Local Coupling Property for Markov Processes with Applications to L\'evy Processes

## TL;DR

This paper introduces the local coupling property for Markov processes, explores its connection with transition probability distributions, and establishes new conditions for absolute continuity in Lévy processes, improving upon existing results.

## Contribution

It defines the local coupling property for Markov processes and links it to distributional properties, providing new criteria for absolute continuity in Lévy processes.

## Key findings

- Local coupling property is equivalent to absolute continuity in Lévy processes.
- Provides a sufficient condition for absolute continuity based on the Lévy measure.
- Results strengthen existing conditions for absolute continuity of Lévy distributions.

## Abstract

In this article, we define the new concept of local coupling property for Markov processes and study its relationship with distributional properties of the transition probability. In the special case of L\'evy processes we show that this property is equivalent to the absolute continuity of the transition probability and also provide a sufficient condition for it in terms of the L\'evy measure. Our result is stronger than existing results for absolute continuity of L\'evy distributions.