The Lévy laplacian and the Yang-Mills equations

Accarde, Luigi di; Gibilisco, Paolo; Волович, И. В.

doi:10.1007/bf03001574

Cited by 18 publications

(31 citation statements)

References 16 publications

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“…Then the Lévy-Laplacian associated with this basis coincides with the Lévy-Laplacian defined in [4,3]. 2 The theorem about equivalence of the Yang-Mills equations and the Lévy-Laplace equation holds for this Laplacian.…”

Section: Introductionmentioning

confidence: 93%

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Stochastic Lévy differential operators and Yang–Mills equations

Volkov

2017

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract:The relationship between the Yang-Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang-Mills equations and the Lévy-Laplace equation for such Laplacian are not equivalent as in the deterministic case. An equation equivalent to the Yang-Mills equations is obtained. The equation contains the stochastic Lévy divergence. It is proved that the Yang-Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of chiral field. This analogue is also derived using Cèsaro averaging.

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Section: Introductionmentioning

confidence: 93%

“…4 The higher order derivatives are defined similarly (see [21]). (About different ways to define the Sobolev classes over Gaussian measures see for example [12] and the references cited therein.…”

Section: Stochastic Parallel Transport and Its Derivativesmentioning

confidence: 99%

Stochastic Lévy differential operators and Yang–Mills equations

Volkov

2017

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…In the paper [2] it was also described a multitude of functionals, on spaces of unbounded operators, corresponding to the series of the so-called exotic Laplacians, whose definition was proposed in [1] (the just defined Levy Laplacian, is the first element of this series). The interest to the study of the Levy Laplacian significantly grew after that, in the paper [1], it was proved that the Euclidean Yang-Mills equations on the n-dimensional Euclidean space, are equivalent to the Laplace equations, corresponding to the Levy Laplacian (in a space of matrix valued functions). In the paper [2] we also described the basic ideas for the use of the Fourier transform (FT) method for obtaining an explicit formula giving the solution of the heat equation, corresponding to the Levy Laplacian.…”

Section: Introductionmentioning

confidence: 99%

Brownian motion generated by the Levy Laplacian

Accardi¹,

Rozelli²,

Smolyanov³

1993

Math Notes

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“…We denote by H 1 (G, F µ (M, U (K))) the set of equivalence classes of continuous cocycles determined by the relation(2).…”

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confidence: 99%

Remarks on the Imprimitivity Theorem for Nonlocally Compact Polish Groups

Fidaleo

2000

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

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The Lévy laplacian and the Yang-Mills equations

Cited by 18 publications

References 16 publications

Stochastic Lévy differential operators and Yang–Mills equations

Stochastic Lévy differential operators and Yang–Mills equations

Brownian motion generated by the Levy Laplacian

Remarks on the Imprimitivity Theorem for Nonlocally Compact Polish Groups

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