Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature

In this paper, we study Perelman's W-entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry-Emery Ricci curvature. Under the assumption that the m-dimensional Bakry-Emery Ricci curvature is bounded from below, we prove an analogue of Perelman's and Ni's entropy formula for the W-entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W-entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry-Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W-entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.