# Log-Harnack Inequalities for Markov Semigroups Generated by Non-Local Gruschin Type Operators

## TL;DR

This paper establishes log-Harnack inequalities for Markov semigroups generated by non-local Gruschin type operators using coupling and regularization techniques, with concrete examples provided.

## Contribution

It introduces a novel approach combining coupling and regularization to derive log-Harnack inequalities for a new class of non-local operators.

## Key findings

- Log-Harnack inequalities are proven for the class of operators.
- The method applies to various concrete examples.
- The approach advances understanding of non-local Markov processes.

## Abstract

Based on coupling in two steps and the regularization approximations of the underlying subordinators, we establish log-Harnack inequalities for Markov semigroups generated by a class of non-local Gruschin type operators. Some concrete examples are also presented.