• Corpus ID: 215768710

Differential Harnack Inequalities on Path Space

@article{Haslhofer2020DifferentialHI,
  title={Differential Harnack Inequalities on Path Space},
  author={Robert Haslhofer and Eva Kopfer and Aaron Naber},
  journal={arXiv: Differential Geometry},
  year={2020}
}
Recall that if $(M^n,g)$ satisfies $\mathrm{Ric}\geq 0$, then the Li-Yau Differential Harnack Inequality tells us for each nonnegative $f:M\to \mathbb{R}^+$, with $f_t$ its heat flow, that $\frac{\Delta f_t}{f_t}-\frac{|\nabla f_t|^2}{f_t^2} +\frac{n}{2t}\geq 0.$ Our main result will be to generalize this to path space $P_xM$ of the manifold. A key point is that instead of considering infinite dimensional gradients and Laplacians on $P_xM$ we will consider a family of finite dimensional… 

References

SHOWING 1-10 OF 17 REFERENCES
Renormalized Differential Geometry on Path Space: Structural Equation, Curvature
Abstract The theory of integration in infinite dimensions is in some sense the backbone of probability theory. On this backbone the stochastic calculus of variations has given rise to the flesh of
Ricci Curvature and Bochner Formulas for Martingales
We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM and develop a formalism to compute evolution equations for martingales on path space. We see
Transformations of Weiner Integrals Under Translations
In his paper on Generalized Harmonic Analysis (in which references to his earlier work are given) N. Wiener [I] defines an average or integral over the space C of all functions x(t) continuous in 0 ?
On the parabolic kernel of the Schrödinger operator
Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x,t)=0 sur une variete riemannienne generale. Introduction. Estimations de gradients. Inegalites de Harnack. Majorations et minorations des
The Foundations Of Differential Geometry
TLDR
The the foundations of differential geometry is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.
An Introduction to the Analysis of Paths on a Riemannian Manifold
Brownian motion in Euclidean space Diffusions in Euclidean space Some addenda, extensions, and refinements Doing it on a manifold, an extrinsic approach More about extrinsic Riemannian geometry
AnL2Estimate for Riemannian Anticipative Stochastic Integrals
Abstract We define a metric and a Markovian connection on the path space of a Riemannian manifold which are different from those introduced in [CM] and prove a corresponding Weitzenbock formula. An L
Stochastic Analysis on the Path Space of a Riemannian Manifold: I. Markovian Stochastic Calculus
Abstract On the Brownian flow of a compact Riemannian manifold, an intrinsic stochastic calculus is defined. This calculus is Markovian. The stochastic calculus of variation on the bundle of
...
...