A Poincaré inequality on loop spaces

Chen, Xin; Li, Xue Mei; Wu, Bo

doi:10.1016/j.jfa.2010.05.006

Cited by 15 publications

(12 citation statements)

References 39 publications

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“…The spectral gap inequality is known to hold for Gaussian measure on R n by L. Gross [20], who also made a conjecture on its validity. The spectral gal inequality has been proven to hold on the hyperbolic space [4], see also [1,17,2,15,11]. A counter example exists [8], see also the more recent articles [21,18].…”

Section: Discussionmentioning

confidence: 99%

On the Semi-Classical Brownian Bridge Measure

Li¹

2016

Preprint

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We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole. IntrodcutionLet M be a finite dimensional smooth connected complete and stochastically complete Riemannian manifold M whose Riemannian distance is denoted by r. By stochastic completeness we mean that its minimal heat kernel satisfies that p t (x, y)dy = 1. Denote by C([0, 1]; M ) the space of continuous curves: σ : [0, 1] → M , a Banach manifold modelled on the Wiener space. A chart containing a path σ is given by a tubular neighbourhood of σ and the coordinate map is induced from the exponential map given by the Levi-Civita connection on the underlying finite dimensional manifold. For x 0 , y 0 ∈ M we denote by C x0 M and C x0,y0 M , respectively, the based and the pinned space of continuous paths over M :

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Section: Discussionmentioning

confidence: 99%

On the Semi-Classical Brownian Bridge Measure

Li¹

2016

Preprint

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“…Hence F is a solution to the Poisson equation. We refer to the following on discussions on irreducibility of Dirichlet forms, Clark-Ocone formula and Poincaré inequalities on path and loop spaces: [1,3,4,6,26,28,30].…”

Section: Exampesmentioning

confidence: 99%

The Stochastic Differential Equation Approach to Analysis on Path Space

Li¹

2011

Interdisciplinary Mathematical Sciences

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“…Aida [4], on the other hand, deduced a Clark-Ocone formula which led to a Logarithmic Sobolev inequality for a modified Dirichlet form, under suitable conditions on the small time asymptotics of the Hessian of the logarithm of the heat kernel of the underlying manifold. Built on that, a Poincaré inequality is shown to hold for the O-U Dirichlet form on the loop space over hyperbolic space, see Chen et al [8].…”

Section: Bmentioning

confidence: 99%

A concrete estimate for the weak Poincaré inequality on loop space

Chen

2010

Probab. Theory Relat. Fields

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The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with a Hilbert space structure and the Ornstein–Uhlenbeck operator d*d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of the underlying manifold. The order of the rate function is s−α for any α > 0

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A Poincaré inequality on loop spaces

Abstract: We show that the Laplacian on the loop space over a class of Riemannian manifolds has a spectral gap. The Laplacian is defined using the Levi-Civita connection, the Brownian bridge measure and the standard Bismut tangent spaces.

Cited by 15 publications

References 39 publications

On the Semi-Classical Brownian Bridge Measure

On the Semi-Classical Brownian Bridge Measure

The Stochastic Differential Equation Approach to Analysis on Path Space

A concrete estimate for the weak Poincaré inequality on loop space

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