Gradient estimates on manifolds using coupling (Q1178829)

Using a probabilistic coupling technique on a complete Riemannian manifold \(M\), an estimate is given for the gradient of solutions to \((1/2\Delta+Z)u=0\). Here \(\Delta\) is the Laplacian and \(Z\) a given vector field. The bound in the gradient estimate depends on the bounds of \(Z\) and on a lower bound on the Ricci curvature of \(M\).