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.

We consider the Itô stochastic differential equation dX t = m j =1 A j (X t ) dw j t + A 0 (X t ) dt on R d . The diffusion coefficients A 1 , . . . , A m are supposed to be in the Sobolev space W 1,p loc (R d ) with p > d, and to have linear growth. For the drift coefficient A 0 , we distinguish two cases: (i) A 0 is a continuous vector field whose distributional divergence δ(A 0 ) with respect to the Gaussian measure γ d exists, (ii) A 0 has Sobolev regularity W 1,p loc for some p > 1. Assume R d exp[λ 0 (|δ(A 0 )| + m j =1 (|δ(A j )| 2 + |∇A j | 2 ))] dγ d < +∞ for some λ 0 > 0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X t ) # γ d admits a density with respect to γ d . In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.

We use martingale methods to give Bismut type derivative formulas for differentials and co-differentials of heat semigroups on forms, and more generally for sections of vector bundles. The formulas are mainly in terms of Weitzenbo ck curvature terms; in most cases derivatives of the curvature are not involved. In particular, our results improve B. K. Driver's formula in (1997, J. Math. Pures Appl. (9) 76, 703 737) for logarithmic derivatives of the heat kernel measure on a Riemannian manifold. Our formulas also include the formulas of K. D. Elworthy and X.-M. Li (1998, C. R. Acad. Sci. Paris Se r. I Math. 327, 87 92). Academic Press

We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain di erentiation formulae for Poisson integrals in the same way as for heat semigroups involving boundary conditions. In particular, our results yield Bismut type representations for the logarithmic derivative of the Poisson kernel on regular domains in Riemannian manifolds corresponding to elliptic PDOs of H ormander type. Such formulae provide a direct approach to gradient estimates for harmonic functions on Riemannian manifolds.

We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.