Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by L = Delta - Delta V.del on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup(x is an element of M)vertical bar del G vertical bar(x) of the Poisson equation -LG = g in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheegcr's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided. (C) 2009 Elsevier Inc. All rights reserved.