Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling

TL;DR

This paper develops coupling-based methods to establish gradient bounds and Liouville properties for hypoelliptic diffusions, leading to new inequalities and estimates that deepen understanding of their behavior.

Contribution

It introduces reverse Bakry-Émery estimates for hypoelliptic operators using coupling, deriving various inequalities and properties not previously established for this class.

Findings

Derived reverse Bakry-Émery estimates for hypoelliptic diffusions

Established Poincaré and logarithmic Sobolev inequalities as consequences

Proved Wang-Harnack inequality, Hamilton's gradient estimate, and Liouville property

Abstract

In this paper, we obtain the reverse Bakry-\'Emery type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincar\'e inequalities and the (right and reverse) logarithmic Sobolev inequalities are presented as consequences of such estimates. Wang-Harnack inequality, Hamilton's gradient estimate and Liouville property are also presented by reverse logarithmic Sobolev inequality.