A characterization of the rate of change of $\Phi$-entropy via an integral form curvature-dimension condition

Abstract

Let $M$ be a compact Riemannian manifold without boundary and $V:M\to \mathbb{R}$ a smooth function. Denote by $P_t$ and $\mathrm{d}\mu =e^V\mathrm{d}x$ the semigroup and symmetric measure of the second order differential operator $L=\Delta +\nabla V\cdot \nabla$. For some suitable convex function $\Phi :\mathcal{I}\to \mathbb{R}$ defined on an interval $\mathcal{I}$, we consider the $\Phi$-entropy of $P_{t}f$ (with respect to $\mu$) for any $f\in C^{\infty }(M,\mathcal{I})$. We show that an integral form curvature-dimension condition is equivalent to an estimate on the rate of change of the $\Phi$-entropy. We also generalize this result to bounded smooth domains of a complete Riemannian manifold.

16 pages. Accepted by Advances in Geometry

References in corpus (2)

• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Generalized Curvature Condition for Subelliptic Diffusion Processes