Startseite Mathematik Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators

  • Hafedh Rguigui EMAIL logo
Veröffentlicht/Copyright: 30. Dezember 2016
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 66 Heft 6

Abstract

In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator ΔEg,Q acting on generalized operators. ΔEg,Q is defined as sum of the extension of the 𝚀𝚆𝙽-Gross Laplacian and the 𝚀𝚆𝙽-conservation operator. It is shown that the operator ΔEg,Q has a representation in terms of the 𝚀𝚆𝙽-derivatives {Dc−,Dc+:c∈N}. The poisson equation is worked out as a non homogeneous Wick differential equation associated to ΔEg,Q.

MSC 2010: Primary 60H40; 46A32; 46F25; 46G20
Keywords: Wick differential equation; 𝚀𝚆𝙽-Gross Laplacian; 𝚀𝚆𝙽-conservation operator; 𝚀𝚆𝙽-Euler operator; Poisson equation
E-mail:

(Communicated by Sylvia Pulmannová)

References

[1] Accardi, L.—Barhoumi, A.—Ji, U. C.: Quantum Laplacians on Generalized Operators on Boson Fock space, Probab. Math. Statist. 31 (2011), 1–24.10.1142/S0219025711004262Suche in Google Scholar

[2] Accardi, L.—Ouerdiane, H.—Smolyanov, O. G.: Lévy Laplacian acting on operators, Russian J. Math. Phys. 10, (2003), 359–380.Suche in Google Scholar

[3] Barhoumi, A.—Lanconelli, A.—Rguigui, H.: 𝚀𝚆𝙽-Convolution operators with application to differential equations, Random Oper. Stoch. Equ. 22 (2014), 10.1515/rose-2014-0019.Suche in Google Scholar

[4] Barhoumi, A.—Ouerdiane, H.—Rguigui, H.: 𝚀𝚆𝙽-Euler Operator And Associated Cauchy problem, Infinite Dimensional Analysis Quantum Probability and Related Topics 15, (2012) 1250004 (20 pages).10.1142/S021902571250004XSuche in Google Scholar

[5] Barhoumi, A.—Ouerdiane, H.—Rguigui, H.: Generalized Euler heat equation, Quantum Probab. White Noise Anal. 25 (2010), 99–116.10.1142/9789814295437_0008Suche in Google Scholar

[6] Barhoumi, A.—Ouerdiane, H.—Rguigui, H.: Stochastic Heat Equation on Algebra of Generalized Functions, Infinite Dimensional Analysis Quantum Probability and Related Topics, Vol. 15, No. 4 (2012) 1250026 (18 pages).10.1142/S0219025712500269Suche in Google Scholar

[7] Ben Chrouda, M.—El Oued, M.—Ouerdiane, H.: Convolution calculus and application to stochastic differential equation, Soochow J. of Mathematics 28, (2002), 375–388.Suche in Google Scholar

[8] Chung, D. M.—Ji, U. C.: Transform on white noise functionals with their application to Cauchy problems, Nagoya Math. J., 147 (1997), 1–23.10.1017/S0027763000006292Suche in Google Scholar

[9] Chung, D. M.—Ji, U. C.: Transformation groups on white noise functionals and their application, Appl. Math. Optim., 37 (1998), 205–223.10.1007/s002459900074Suche in Google Scholar

[10] Chung, D. M.—Ji, U. C.—Obata, N.: Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), 241–272.10.1142/S0129055X0200117XSuche in Google Scholar

[11] Gannoun, G.—Hachaichi, R.—Ouerdiane, H.—Rezgi, A.: Un théorème de dualité entre espace de fonction holomorphes à croissance exponentielle, J. Funct. Anal. 171 (2000), 1–14.10.1006/jfan.1999.3518Suche in Google Scholar

[12] Gross, L.: Abstract Wiener spaces, Proc. 5-th Berkeley Symp. Math. Stat. Probab.2 (1967), 31–42.Suche in Google Scholar

[13] Hida, T.: A role of the Lévy Laplacian in the causal calculus of generalized white noise functionals, ”Stoch. Proc., A Festschrift in Honour of G. Kallianpur” (S. Cambanis et al. Eds.) Springer-Verlag 1992.10.1007/978-1-4615-7909-0_16Suche in Google Scholar

[14] Hida, T.—Ikeda, N.: Analysis on Hilbert space with reproducing kernels arising from multiple Wiener integrals, Proc. Fifth Berkeley Symp. Math. Stat. Prob., Vol. II, Part 1 (1965), 117–143.10.1142/9789812794611_0009Suche in Google Scholar

[15] Horrigue, S.—Ouerdiane, H.: Quantum heat equation with Quantum K-Gross Laplacian: Solutions and Integral representation, Quantum Probab. White Noise Anal. 25, (2010), 185–202.10.1142/9789814295437_0013Suche in Google Scholar

[16] Ji, U. C.: Integral kernel operators on regular generalized white noise functions, Bull. Korean Math. Soc. 37 (2000), 601–618.Suche in Google Scholar

[17] Ji, U. C.: Quantum Extensions of Fourier-Gauss and Fourier-Mehler Transforms, J. Korean Math. Soc. 45 (2008), 1785–1801.10.4134/JKMS.2008.45.6.1785Suche in Google Scholar

[18] Ji, U. C.—Obata, N.: Generalized white noise operator fields and quantum white noise derivatives, Séminaires & Congrès 16 (2007), 17–33.Suche in Google Scholar

[19] Ji, U. C.—Obata, N.: Annihilation-derivative, creation-derivative and representation of quantum martingales, Commun. Math. Phys. 286 (2009), 751–775.10.1007/s00220-008-0702-3Suche in Google Scholar

[20] Ji, U. C.—Obata, N.: Quantum stochastic integral representations of Fock space operators, Stochastics: An International Journal of Probability and Stochastics Processes, Vol. 81, (2009), 367–384.10.1080/17442500902919645Suche in Google Scholar

[21] Ji, U. C.—Obata, N.: Quantum White Noise Derivatives and Associted Differential Equation for White Noise Operator, Quantum Probab. White Noise Anal. 25 (2010), 42–54.10.1142/9789814295437_0004Suche in Google Scholar

[22] Ji, U. C.—Obata, N.: Implementation problem for the canonical commutation relation in terms of quantum white noise derivatives, J. Math. Phys. 51, 123507 (2010).10.1063/1.3516477Suche in Google Scholar

[23] Ji, U. C.—Obata, N.—Ouerdiane, H.: Analytic characterization of generalized Fock space operators as two-variable entire function with growth condition, Infinite Dimensional Analysis Quantum Probability and Related Topics 5 (2002), 395–407.10.1142/S0219025702000912Suche in Google Scholar

[24] Kuo, H. H.: White noise distribution theory, CRC press, Boca Raton 1996.Suche in Google Scholar

[25] Meyer, P. A.—Yan, J. A.: Distributions sur l’espace de Wiener (suite)., Séminaire de Probabilités XXIII, J. Azéma, P.A. mayer and M. Yor (eds.). Springer, Berlin, Heidelberg, New York, 1989.10.1007/BFb0083955Suche in Google Scholar

[26] Obata, N.: An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), 421–445.10.2969/jmsj/04530421Suche in Google Scholar

[27] Obata, N.: White noise calculus and Fock spaces, Lecture Notes in Math. 1577, Spriger-Verlag 1994.10.1007/BFb0073952Suche in Google Scholar

[28] Obata, N.: Quantum white noise calculus based on nuclear algebras of entire function, Trends in Infinite Dimensional Analysis and Quantum Probability (Kyoto 2001), RIMS No. 1278, 130–157.Suche in Google Scholar

[29] Ouerdiane, H.—Rguigui, H.: 𝚀𝚆𝙽-Conservation Operator And Associated Wick Differential Equation, Communication on stochastic analysis 6 (2012), 437–450.10.31390/cosa.6.3.06Suche in Google Scholar

[30] Piech, M. A.: Parabolic equations associated with the number operator, Trans. Amer. Math. Soc. 194 (1974), 213–222.10.1090/S0002-9947-1974-0350231-3Suche in Google Scholar

Received: 2014-6-9
Accepted: 2014-11-11
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
https://doi.org/10.1515/ms-2016-0238
Schlagwörter für diesen Artikel
Wick differential equation; 𝚀𝚆𝙽-Gross Laplacian; 𝚀𝚆𝙽-conservation operator; 𝚀𝚆𝙽-Euler operator; Poisson equation

Artikel in diesem Heft

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2016-0238/html
Button zum nach oben scrollen