Levy Laplacian Research Articles

Applications of Lévy Differential Operators in the Theory of Gauge Fields

This paper is a survey of results on the relationship between gauge fields and infinitedimensional equations for parallel transport that contain the Levy Laplacian or the divergence associated with this Laplacian. Also we analyze the deterministic case where parallel transports are operator-valued functionals on the space of curves and the case of the Malliavin calculus where (stochastic) parallel transports are operator-valued Wiener functionals.

Levy Laplacian on Manifold and Yang—Mills Heat Flow

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang-Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Laplacian on the infinite dimensional manifold.

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Levy 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 Levy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Levy Laplacian. Another approach to define the Levy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Levy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Levy Laplacians, which is not the classical Levy 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 Levy Laplacian in the stochastic analysis is connected to the gauge fields.

Levy Laplacian on a Four-Dimensional Riemannian Manifold

It is known that the Yang-Mills equations for a connection are equivalent to the Levy-Laplace equation for the parallel transport associated with this connection (see the paper [4] by L. Accardi, P. Gibilisco and I. V. Volovich). The Levy-Laplace equation is the Laplace equation for Levy Laplacian, which can be defined as the Cesaro mean of the second-order directional derivatives along the vectors from the orthonormal basis of some Hilbert space. The author’s work [11] has proved for the case of a four-dimensional Euclidean space, that with a certain choice of the orthonormal basis, the Laplace-Levy equation for a parallel transport becomes equivalent to the self-duality equations for a connection. A connection that is a solution to the self-duality equations is called an instanton and is a solution to the Yang-Mills equations. In the paper we define the Levy Laplacian for the case of a four-dimensional Riemannian manifold. This operator is a generalization of both the Levy Laplacian, introduced by the author in [11], and the Levy Laplacian, introduced in [3] by L. Accardi and О. G. Smolyanov for a Riemannian manifold. In the paper we consider the case of a line bundle over a four-dimensional Riemannian manifold and Maxwell's equations (the commutative case of the Yang-Mills equations). We find the conditions, which imply that from the fact that the parallel transport is a harmonic functional for the introduced Levy Laplacian follows that the appropriate connection is a solution to the self-duality equations. In addition, the paper considers the relationship of the introduced Levy Laplacian and the Laplace-Beltrami operator on a manifold. It can be expected that the results of the paper can be generalized to the non-commutative case of Yang-Mills fields.

Stochastic Levy Laplacian and d'Alambertian and Maxwell's Equations

Stochastic Levy Laplacian and d'Alambertian and Maxwell's Equations

Lévy Laplacians and instantons

We describe dual and antidual solutions of the Yang–Mills equations by means of L´evy Laplacians. To this end, we introduce a class of L´evy Laplacians parameterized by the choice of a curve in the group SO(4). Two approaches are used to define such Laplacians: (i) the Levy Laplacian can be defined as an integral functional defined by a curve in SO(4) and a special form of the second-order derivative, or (ii) the Levy Laplacian can be defined as the Cesaro mean of second-order derivatives along vectors from the orthonormal basis constructed by such a curve. We prove that under some conditions imposed on the curve generating the Levy Laplacian, a connection in the trivial vector bundle with base R4 is an instanton (or an anti-instanton) if and only if the parallel transport generated by the connection is harmonic for such a Levy Laplacian.

Quantum probability and Levy Laplacians

The formula ΔL = limɛ→0 ∫∥s–t∥<ɛbsbtdsdt for the Levy Laplacian is obtained, where bt stands for an annihilation process. The formula is extended to some generalizations of the Levy Laplacian.

Boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian resolved with respect to the derivative

We present the solutions of boundary-value and initial boundary-value problems for a nonlinear parabolic equation with Levy Laplacian ∆ L resolved with respect to the derivative $$ \frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {U\left( {t,x} \right),{\Delta_L}U\left( {t,x} \right)} \right) $$ in fundamental domains of a Hilbert space.

The Cauchy problem for the wave equation with Lévy Laplacian

We present the solution of the Cauchy problem (the initial-value problem in the whole space) for the wave equation with infinite-dimensional Levy Laplacian ΔL, $$ \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} = \Delta _L U(t,x) $$ in two function classes, the Shilov class and the Gâteaux class.

A stochastic process associated with the weighted white noise differentiation

In this paper, we give a relationship between the weighted white noise dierentiation and the Levy Laplacian introducing an operator changing the white noise ˙ B(t) by ˙ B(t) 2 . In addition, we give an infinite dimensional stochastic process generated by the weighted white noise dierentiation and a relationship between the stochastic process and the Levy Laplacian.

Generalized Lévy Laplacians and Cesàro means

In this paper, we show that the composition of the generalized Levy Laplacian of order p (which is gener- ated by the corresponding Cesaro mean) and a certain linear transformation of the domain of the function to which it is applied is proportional to the generalized Levy Laplacian of smaller or larger order; this makes it possible to transform the generalized Levy Laplacian of any order into the classical Levy Laplacian. The con- struction is based on an expression of generalized Cesaro means in terms of ordinary ones, which is also suggested in this paper. By virtue of results of (5), this implies that all results on the classical Levy Laplacian have analogues for the generalized Levy Laplacian. In particular, we consider a relationship between the gen- eralized Levy Laplacian and quantum random pro- cesses.

A Gauss-Poisson Correspondence and the Lévy Laplacian

In this paper we present recent results on infinite dimensional Laplacians. In particular, by introducing the operator which transfers regular white noise functionals to functionals of exponential white noise, we give a relationship between an infinite dimensional Fourier-Mehler transform and the Levy Laplacian. The operator implies a Gauss-Poisson correspondence if we consider the Levy Laplacian acting on multiple Wiener integrals by some Levy process. We also give an infinite dimensional random field associated with the Levy Laplacian.

Modified Lévy Laplacians

An infinite family of Levy-type Laplacians is introduced and investigated (its elements are both the classical Levy Laplacians and some of their modifications), and a relationship between these operators and some quantum random processes is described.

Classical and nonclassical levy Laplacians

A general method for defining and studying operators introduced by Paul Levy referred as classical Levy Laplacians and their modifications referred as nonclassical Laplacians, which makes its possible to extend results on Levy Laplacians to nonclassical Laplacians, has been discussed. An infinite family of Laplacians, whose elements are classical Laplacians and the non-classical Laplacians related to the Yang-Mills equations, has been defined and the relationship between these Laplacians and quantum random process has been described. A quantum random process is a function defined on a part of the real line and taking values in some space of operators. Volterra Laplacians have also been considered and analogies between them and Levy Laplacians have been found.

Infinite dimensional Laplacians on a Lévy-Gelfand triple

We introduce a new Hilbert space HL, called Levy{Hilbert space, arising from the LLaplacian. This Hilbert space is used to construct a new Gel'fand triple called Levy{Gel'fand triple, which exhibits the essentially inflnite dimensional character of the Levy Laplacian. By using the fact that the usual trace and the Ltrace (being associated with an appropriate orthonormal system) coincide on the Lert space, we prove a theorem asserting that the Levy Laplacian and the Gross Laplacian coincide on the space of Ltest functions as continuous linear operators.

Feynman formulas for evolution equations with Levy Laplacians on infinite-dimensional manifolds

A Feynman formula is a representation of the solution to the Cauchy problem for an evolution partial differential (or pseudodifferential) equation in terms of the limit of a sequence of multiple integrals with multiplicities tending to infinity. The integrands are products of the initial condition and Gaussian (or complex Gaussian) exponentials 1 [5]. In this paper, we obtain Feynman formulas for the solutions to the Cauchy problems for the Schrodinger equation and the heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold. The definition of the Levi Laplacian acting on functions on such a manifold is obtained by combining the methods of papers [3] and [7]. In the former, Levi Laplacians in the space of functions on an infinitedimensional vector space were considered, and in the latter, Volterra Laplacians in the space of functions on the above infinite-dimensional manifold were examined. This definition of a Levi Laplacian is equivalent to that given in [2], but it is better adapted for derivation of Feynman formulas. The main idea of the proof of the central result of this paper is reducing the derivation of Feynman formu1 For the heat equation, these multiple integrals coincide with integrals being finite-dimensional approximations to integrals with respect to the Wiener measure. For the Schrodinger equation, such integrals coincide with those used in the definition (which goes back to Feynman himself) of sequential Feynman path integrals. Therefore, the limits of multiple integrals in the Feynman formulas are integrals with respect to the Wiener measure in the former case and (sequential) Feynman path integrals in the latter case, and in both cases, the Feynman‐Kac-type formulas are consequences of the Feynman formulas being discussed. las for equations on a manifold to the derivation of similar formulas for equations on a vector space. For equations on finite-dimensional manifolds (containing the usual finite-dimensional Laplacians), this approach was suggested in [4] and developed in [6]. The Riemannian manifold under consideration was embedded in a suitable Euclidean space (this can always be done by the Nash theorem), and the technique of surface measures developed in [4, 7] was applied. An essential point in the proof was the application of the Chernoff formula (generalizing the Trotter formula), which is related to obtaining representations of solutions to evolution equations on manifolds (and to representations of solutions to equations on vector spaces in terms of path integrals in the phase space [5]) in the same way as the Trotter formula is related to representations of the solution to the simplest Schrodinger equation with potential in terms of path integrals in the configuration space. The remark made in the footnote means that the results obtained in this paper contain the construction

Lévy--Laplace Operators in Functional Rigged Hilbert Spaces

New results on analytical properties of the Levy–Laplace operator (the Levy Laplacian) and the semigroups generated by this operator are established in this paper. To this end, we use the methods developed earlier for the Levy–Volterra operators (all necessary definitions are given below); the corresponding results are similar in both cases. The analogy arises because Hilbert space is replaced by a rigged Hilbert space; in fact, Hilbert space corresponds to a degenerate case (cf. [1, 2]): it is just in this case that the Levy Laplacian possesses “unusual” properties (cf. [3, 4]). Our approach develops the techniques proposed in [1, 5–8].

A Class of generalized Lévy Laplacians in infinite dimensional calculus

A class of generalized Levy Laplacians which contain as a special case the ordinary Levy Laplacian are considered. Topics such as limit average of the second order functional derivative with respect to a certain equally dense (uniformly bounded) orthonormal base, the relations with Kuo’s Fourier transform and other infinite dimensional Laplacians are studied.

On a nonlinear equation unsolved with respect to the levy laplacian

We propose a method for the solution of the nonlinear equationf(U(x),ΔU(x))=F(x) (ΔL is an infinite-dimensional Laplacian, ΔLU(x)=γ, γ≠0) unsolved with respect to the infinite-dimensional Laplacian, and for the solution of the Dirichlet problem for this equation.

A Note on the Levy Laplacian

1 Introduction The Levy Laplacian was first introduced by P. Levy in studying functionals on L~2 [0, 1] and has been investigated by many authors. In the white noise analysis setting Hida first defined Levy Laplacian Δ_L via the second variation of a U-functional and proved that Δ_L annihilates functionals of square integrable (cf. Refs. [3, 4]). InRef. [3], Hida and Sait proved the following formula: Δ_L(?)=-(?)- (Δ_LF)~^, where F is Kuo’s Fourier transform of F. In Ref. [4], according to the original idea of P. Levy a definition of the Levy Laplacian was proposed. In the present note we will give a new ex-