Generalized Curvature Condition for Subelliptic Diffusion Processes

TL;DR

This paper introduces a generalized curvature condition that leads to derivative inequalities for subelliptic diffusion processes, enabling new bounds and inequalities such as Harnack, Poincaré, and log-Sobolev inequalities.

Contribution

It develops a broad curvature framework that extends and improves existing inequalities for subelliptic diffusion semigroups and associated Dirichlet forms.

Findings

Establishment of derivative inequalities under the new curvature condition

Derivation of Harnack, cost-entropy, and cost-variance inequalities

Generalization and improvement of previous results in the literature

Abstract

By using a general version of curvature condition, derivative inequalities are established for a large class of subelliptic diffusion semigroups. As applications, the Harnack/cost-entropy/cost-variance inequalities for the diffusion semigroups, and the Poincar\'e/log-Sobolev inequalities for the associated Dirichlet forms in the symmetric case, are derived. Our results largely generalize and partly improve the corresponding ones obtained recently in [BB].