Higher order multi-dimensional extensions of Cesàro theorem

Accardi, Luigi; Ji, Un Cig; Saitô, Kimiaki

doi:10.1142/s0219025715500307

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…[5,7] for the definition), which are studied in the white noise analysis (see Ref. [4] and references therein), can be represented as non-classical Lévy Laplacians (see Refs. [26,28]).…”

Section: Remark 2 the Non-classical Lévy Laplacian Is A Generalizatimentioning

confidence: 99%

Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

Volkov

2019

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

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We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang-Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang-Mills equations. This system is an analogue of the equation of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang-Mills-Higgs equations and the Yang-Mills-Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

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“…[5,7] for the definition), which are studied in the white noise analysis (see Ref. [4] and references therein), can be represented as non-classical Lévy Laplacians (see Refs. [26,28]).…”

Section: Remark 2 the Non-classical Lévy Laplacian Is A Generalizatimentioning

confidence: 99%

Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

Volkov

2019

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

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“…Another approach to representation of the Yang-Mills action functional by the stochastic parallel transport has been used in [11]. For the recent development in the study of Lévy Laplacians see [5,6,26].…”

Section: Introductionmentioning

confidence: 99%

“…If y 1 and y 2 are semi-martingales dy 1 • dy 2 denotes the differential of the quadratic covariation of y 1 and y 2 6. The symbol Ind s≤t denotes the indicator of the set {(s, t) : s ≤ t}…”

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confidence: 99%

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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“…We end by noticing that particular cases of averages considered here appear in Section 5 of [4] (see also [3]), where also several continuous versions of averages are investigated.…”

Section: Introductionmentioning

confidence: 99%