A probabilistic approach to the Yang–Mills heat equation

Arnaudon, Marc; Bauer, Robert O.; Thalmaier, Anton

doi:10.1016/s0021-7824(02)01254-0

Cited by 19 publications

(46 citation statements)

References 25 publications

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“…Theorems 3.1, 3.2, and 3.3 provide three different sets of conditions ensuring that R ∇(t) does not blow up. Each of these sets corresponds to a certain range of dimensions of M. A similar trichotomy occurs on closed manifolds; see, for instance, [2]. However, the difference in the geometric assumptions that was discussed in the previous paragraph is not observed in this case.…”

Section: Introductionmentioning

confidence: 88%

“…The proofs of Theorems 3.1, 3.2, and 3.3 rely on the probabilistic technique developed in [2]. The origin of this technique lies in the theory of harmonic maps; see [39].…”

Section: Introductionmentioning

confidence: 99%

“…The pivotal stochastic process in our considerations is a reflecting Brownian motion on the manifold M. Let us mention that the probabilistic approach to Yang-Mills theory was investigated rather extensively. The paper [2] contains a series of results and a list of references on the subject.…”

Section: Introductionmentioning

confidence: 99%

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The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

Pulemotov

2008

Journal of Functional Analysis

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The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution ∇(t) of the Yang-Mills heat equation in a vector bundle over M. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of ∇(t) does not blow up if the dimension of M is less than 4 or if the initial energy of ∇(t) is sufficiently small.

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Section: Introductionmentioning

confidence: 88%

“…The proofs of Theorems 3.1, 3.2, and 3.3 rely on the probabilistic technique developed in [2]. The origin of this technique lies in the theory of harmonic maps; see [39].…”

Section: Introductionmentioning

confidence: 99%

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The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

Pulemotov

2008

Journal of Functional Analysis

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“…A slight modification in [4] of this construction yields a martingale representation of the heat equation for Yang-Mills connections. A monotonicity formula for the quadratic variation of the martingale is derived in [4], as well as nonexplosion criteria for the heat equation involving the quadratic variation of the martingale.…”

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

confidence: 99%

“…A monotonicity formula for the quadratic variation of the martingale is derived in [4], as well as nonexplosion criteria for the heat equation involving the quadratic variation of the martingale.…”

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

confidence: 99%

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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The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary

Charalambous

Gross²

2012

Commun. Math. Phys.

114

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Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H 1 . Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic non-linear equation. We use a technique of Donaldson and Sadun to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.

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A probabilistic approach to the Yang–Mills heat equation

Cited by 19 publications

References 25 publications

The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

Yang--Mills fields and random holonomy along Brownian bridges

The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary

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