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Remarks on spectral gaps on the Riemannian path space
Shizan Fang,Bo Wu +1 more
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In this article, the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold is investigated.Abstract:
In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.read more
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Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms
TL;DR: In this paper, the authors proved the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits the Wiener (Brownian bridge) measure on the Rieman manifold.
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Characterization of Pinched Ricci Curvature by Functional Inequalities
Li Juan Cheng,Anton Thalmaier +1 more
TL;DR: In this paper, functional inequalities for diffusion semigroups on Riemannian manifolds are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, and log-Sobolev inequalities.
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Pointwise Characterizations of Curvature and Second Fundamental Form on Riemannian Manifolds
Feng-Yu Wang,Feng-Yu Wang,Bo Wu +2 more
TL;DR: In this paper, pointwise characterizations are presented for the Bakry-Emery curvature of a complete Riemannian manifold possibly with a boundary, and the second fundamental form of the curvature if exists.
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Characterizations of the Upper Bound of Bakry–Emery Curvature
TL;DR: In this paper, Wang et al. presented characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on the path space.
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Spectral gap on Riemannian path space over static and evolving manifolds
Li Juan Cheng,Anton Thalmaier +1 more
TL;DR: In this article, the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature was investigated.
References
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A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold
TL;DR: In this paper, it was shown that σ(t) has the quasi-invariance property: the law (σ(t ∗ ν ) with respect to the Wiener measure (ν) is equivalent to ν for all tϵ.
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Formulae for the derivatives of heat semigroups
K. D. Elworthy,Xue-Mei Li +1 more
TL;DR: In this paper, a martingale method was used to show a differentiation formula for derivatives for the derivatives of a heat equation on differential forms and a second order formula for solutions of heat equations on manifolds.
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A logarithmic Sobolev form of the Li-Yau parabolic inequality
Dominique Bakry,Michel Ledoux +1 more
TL;DR: In this paper, a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators was presented, which contains and improves upon the Li-Yau parabolic inequality.
Book
An Introduction to the Analysis of Paths on a Riemannian Manifold
TL;DR: Brownian motion in Euclidean space Diffusions and Brownian paths in Riemannian geometry are discussed in this article, where the authors propose an extrinsic approach to do it on a manifold.
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