Li-Yau Type Gradient Estimates and Harnack Inequalities by Stochastic Analysis

In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local estimates in the sense that the gradient bound at given point depends only on universal constants and the geometry of the Riemannian manifold locally about this point

The differentiation of hypoelliptic diffusion semigroups

Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. Emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with boundary conditions, as well as with finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally some extensions to the nonlinear setting of harmonic mappings are discussed

Concentration of the Brownian bridge on Cartan-Hadamard manifolds with pinched negative sectional curvature

We study the rate of concentration of a Brownian bridge in time one around the corresponding geodesical segment on a Cartan-Hadamard manifold with pinched negative sectional curvature, when the distance between the two extremities tends to infinity. This improves on previous results by A. Eberle, and one of us. Along the way, we derive a new asymptotic estimate for the logarithmic derivative of the heat kernel on such manifolds, in bounded time and with one space parameter tending to infinity, which can be viewed as a counterpart to Bismut's asymptotic formula in small time

Application of Stochastic Flows to the Sticky Brownian Motion Equation

We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation