134 research outputs found
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
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