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 ${\rm d}\mu=e^V\,{\rm d} x$ 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.