Horizontal martingales in vector bundles

Arnaudon, Marc; Thalmaier, Anton

doi:10.1007/978-3-540-36107-7_22

Cited by 8 publications

(17 citation statements)

References 9 publications

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“…Bauer [6], establishes a characterization of Yang-Mills connections in terms of parallel transport along perturbed Brownian motion: the covariant derivative of parallel transport with respect to variations induced by the flow of a gradient-type vector field on the base manifold, parallel transported back to the starting point, is a martingale if and only if the connection is Yang-Mills. In [3] the present authors give a similar characterization: …”

supporting

confidence: 66%

“…These characterizations are consequences of Theorem 3.1, which has been established in [3]. Theorem 3.1 gives the asymptotic expansion in a at a = 0 of the parallel transport W (a) = W (a, (u 1 , u 2 )) in E along a Brownian bridge X(a) starting from exp x 0 (au 1 ) and ending at exp x 0 (au 2 ) at time 1, with quadratic variation a 2 mt, where…”

Section: Moreover the Quadratic Variation S Of W −1 ∇W Is Given Bymentioning

confidence: 68%

“…The following theorem, which has been proved in [3], describes how covariant derivatives with respect to a and t commute. We write …”

Section: Moreover the Quadratic Variation S Of W −1 ∇W Is Given Bymentioning

confidence: 99%

“…Now differentiating (3.1) with respect to a by means of Theorem 2.2 in [3] and taking the covariant derivative according to [3], (4.7), and (1.9) above, we get (using that ∇ is torsion-free)…”

Section: Thus (311) Leads Tomentioning

confidence: 99%

“…Let R be a smooth section of T * M ⊗ T * M ⊗ End (E) over M. (see, e.g., [3], Section 5). Our main example is E = T M and R(u, v)w = R(w, u)v, which gives trR = Ric .…”

Section: Thus (311) Leads Tomentioning

confidence: 99%

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Yang--Mills fields and random holonomy along Brownian bridges

Arnaudon¹,

Thalmaier²

2003

Ann. Probab.

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We characterize Yang-Mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of Brownian bridges on the base manifold. In particular, we prove that a connection in a vector bundle E is Yang-Mills if and only if the covariant derivative of parallel transport along Brownian bridges (in the direction of their drift) is a local martingale, when transported back to the starting point. We present a Taylor expansion up to order 3 for stochastic parallel transport in E along small rescaled Brownian bridges and prove that the connection in E is Yang-Mills if and only if all drift terms in the expansion (up to order 3) vanish or, equivalently, if and only if the average rotation of parallel transport along small bridges and loops is of order 4. Introduction.This article is concerned with the characterization of YangMills connections in a vector bundle E over a compact Riemannian manifold M in terms of stochastic parallel transport along Brownian bridges. Recall that YangMills connections in a vector bundle E with a metric over M are the critical points of the following functional (the so-called Yang-Mills action):

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supporting

confidence: 66%

Section: Moreover the Quadratic Variation S Of W −1 ∇W Is Given Bymentioning

confidence: 68%

“…The following theorem, which has been proved in [3], describes how covariant derivatives with respect to a and t commute. We write …”

Section: Moreover the Quadratic Variation S Of W −1 ∇W Is Given Bymentioning

confidence: 99%

Section: Thus (311) Leads Tomentioning

confidence: 99%

“…Let R be a smooth section of T * M ⊗ T * M ⊗ End (E) over M. (see, e.g., [3], Section 5). Our main example is E = T M and R(u, v)w = R(w, u)v, which gives trR = Ric .…”

Section: Thus (311) Leads Tomentioning

confidence: 99%

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Yang--Mills fields and random holonomy along Brownian bridges

Arnaudon¹,

Thalmaier²

2003

Ann. Probab.

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A Note on Stochastic Calculus in Vector Bundles

Catuogno

Ledesma

Ruffino

2013

Lecture Notes in Mathematics

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Horizontal Diffusion in C 1 Path Space

Arnaudon

Coulibaly

Thalmaier

2010

Séminaire De Probabilités XLIII

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We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.

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Horizontal martingales in vector bundles

Cited by 8 publications

References 9 publications

Yang--Mills fields and random holonomy along Brownian bridges

Yang--Mills fields and random holonomy along Brownian bridges

A Note on Stochastic Calculus in Vector Bundles

Horizontal Diffusion in C 1 Path Space

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