Generalized Lévy Laplacians and Cesàro means

Accardi, Luigi; Smolyanov, O. G.

doi:10.1134/s106456240901027x

Cited by 10 publications

(6 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…to E. The generalized Lévy-Laplacian ∆ l L of order l ≥ 0 is the second-order linear differential operator associated with S l which is defined by S l (F ) = C l ((F e n , e n )), F ∈ L(E, E * ). 7,10 This definition coincides for l = 0 with that of the LaplaceVolterra operator and for l = 1 with that of the classical Lévy-Laplacian. We also consider nonclassical Lévy-Laplacians ∆ L R associated with linear operator R : span{e n : n ∈ N} → E which we define as second-order linear differential operators D SR , where S R (F ) = C 1 ((FRe n , Re n )), F ∈ L(E, E * ).…”

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Hierarchy of Lévy–laplacians and Quantum Stochastic Processes

Volkov

2013

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

View full text Add to dashboard Cite

Communicated by O. SmolyanovWe consider a family of infinite dimensional Laplace operators which contains the classical Lévy-Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy-Laplacian, the following formula is proved: ∆ L = lim ε→0 R s−t <ε bsbtdsdt, where bt is the annihilation process.

show abstract

Section: Introductionmentioning

confidence: 89%

“…2, we follow Ref. 10 and consider the connection between generalized and nonclassical Lévy-Laplacians. In Sec.…”

Section: Introductionmentioning

confidence: 99%

Hierarchy of Lévy–laplacians and Quantum Stochastic Processes

Volkov

2013

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

View full text Add to dashboard Cite

show abstract

“…Remark 1. For the first time formula (4) was used for study of the exotic Lévy Laplacians in [9]. The Lévy Laplacians ∆ {en},s L are the particular case of nonclassical Lévy Laplacians (see [8,30]).…”

Section: Lévy Laplaciansmentioning

confidence: 99%

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Volkov

2018

Proc. Steklov Inst. Math.

View full text Add to dashboard Cite

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.

show abstract

“…This generalization was achieved in successive steps in increasing order of generality: the first result, obtained in [10], concerned sequences (as in the original Cesàro theorem) and means of integer order. The second result in [10] concerns the converse of the first one: this seems to be a new type of Cesàro theorems, not previously considered in the literature. Both results played a crucial role in the construction, given in [4] of a similarity relation among exotic Laplacians of order ≥ 1.…”

Section: Introductionmentioning

confidence: 99%