Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

TL;DR

This paper establishes differential Harnack inequalities and Perelman type entropy formulas for subelliptic operators under generalized curvature conditions, advancing understanding of their heat equation solutions and entropy behavior.

Contribution

It introduces new differential Harnack inequalities and entropy formulas for subelliptic operators, extending classical results to a broader geometric setting.

Findings

Derived differential Harnack inequalities for subelliptic Schrödinger equations

Established monotonicity of Perelman type entropy for these operators

Obtained parabolic Harnack inequalities as applications

Abstract

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Sch\"odinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality. Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity.