Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators

TL;DR

This paper investigates the convergence to equilibrium of hypoelliptic, non-symmetric Ornstein-Uhlenbeck operators using a generalized curvature-dimension inequality, with applications to operators on two-step Carnot groups.

Contribution

It introduces a new curvature-dimension inequality tailored for non-symmetric subelliptic operators and establishes convergence results in both $L^2$ and entropic frameworks.

Findings

Proves convergence to equilibrium in $L^2$ and entropic senses.

Applies results to Ornstein-Uhlenbeck operators on two-step Carnot groups.

Develops a framework for non-symmetric hypoelliptic operators.

Abstract

We study a generalized curvature dimension inequality which is suitable for subelliptic Ornstein-Uhlenbeck type operators and deduce convergence to equilibrium in the $L^2$ and entropic sense. The main difficulty is that the operators we consider may not be symmetric. Our results apply in particular to Ornstein-Uhlenbeck operators on two-step Carnot groups.