Stochastic Lévy differential operators and Yang–Mills equations

Proc. Steklov Inst. Math.

2015

Dedicated to the memory of Alexander A. Belyaev Abstract: The equivalence of the anti-selfduality Yang-Mills equations on the 4dimensional orientable Riemannian manifold and Laplace equations for some infinite dimensional Laplacians is proved. A class of modificated Lévy Laplacians parameterized by the choice of a curve in the group SO(4) is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang-Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modificated Levy Laplacians from this class.

“…The last equality holds due to tr(P L (L W (t))Q S − (t)) = tr(P R (L W (t))Q S + (t)) = 0. Equality (32) implies the statement of the proposition.…”

Section: Definition 2 the Levy Trace Trmentioning

confidence: 70%

Section: Intoductionmentioning

confidence: 99%

Lévy Laplacians and instantons

Proc. Steklov Inst. Math.

2015

“…With the help of such analogue it would be possible to develop the results of the paper Ref. [30], where the stochastic parallel transport was considered as a general chiral field. Also it would be interesting to investigate the connection between the stochastic Lévy Laplacians introduced in Ref.…”

Section: Discussionmentioning

confidence: 99%

“…In Refs. [30,31] the Levy Laplacian defined as the Chesaro mean of the directional derivatives in the stochastic case was studied. 2 It was shown that, unlike the deterministic case, the equivalence of the Yang-Mills equations and the Levy-Laplace equation is not valid for such Laplacian.…”

Section: Introductionmentioning

confidence: 99%

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

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

2019

Self Cite

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.

“…In the deterministic planar case, it has been shown in work [29] that this is indeed the case (see also [7]). In the author's work [31], using the Malliavin calculus, the Lévy Laplacian, defined as the Cesàro mean of second partial derivatives, has been introduced on the Sobolev space over the Wiener measure and its relation to gauge fields has been studied. It should be noted that, unlike the deterministic case, the Lévy Laplacians on the Sobolev spaces over the Wiener measure, introduced in the works [22] and [31], operate in different ways.…”

Section: Introductionmentioning

confidence: 99%

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Proc. Steklov Inst. Math.

2018

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev-Schwartz distributions over the Wiener measure (the Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to define the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, which is not the classical Lévy Laplacian, under the imbedding of the Sobolev space over the Wiener measure into the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected to the gauge fields.