Hamilton differential Harnack inequality and $W$-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K,m)$-condition, where $m\in [n,\mathrm{\infty })$ and $K\ge 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K,m)$-condition and on compact manifolds equipped with $(-K,m)$-super Ricci flows.