Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimensionfree Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by −c(1 + ρ 2 o), where c > 0 is a constant and ρ o is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.

Derivative formulae for heat semigroups are used to give gradient estimates for harmonic functions on regular domains in Riemannian manifolds. This probabilistic method provides an alternative to coupling techniques, as introduced by Cranston, and allows us to improve some known estimates. We discuss two slightly different ways to exploit derivative formulae where each one should be interesting by itself.1998 Academic Press

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived. The main resultLet M be a non-compact complete connected Riemannian manifold, and P t be the Dirichlet diffusion semigroup generated by L = ∆ + ∇V for some C 2 function V . We intend to establish reasonable gradient estimates and Harnack type inequalities for P t . In case that Ric − Hess V is bounded below, a dimension-free Harnack inequality was established in [14] which, according $ Supported in part by WIMICS, NNSFC(10721091) and the 973-Project.

Given an n-dimensional compact manifold M, endowed with a family of Riemannian metrics g(t), a Brownian motion depending on the deformation of the manifold (via the family g(t) of metrics) is defined. This tool enables a probabilistic view of certain geometric flows (e.g. Ricci flow, mean curvature flow). In particular, we give a martingale representation formula for a non-linear PDE over M, as well as a Bismut type formula for a geometric quantity which evolves under this flow. As application we present a gradient control formula for the heat equation over (M, g(t)) and a characterization of the Ricci flow in terms of the damped parallel transport.