Abstract
We construct an infinite dimensional analysis with respect to non-Gaussian measures of fractional Gamma type which we call fractional Gamma noise measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using generalized Appell polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the fractional Gamma measures. Finally, we gives some new properties of the kernels expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions and we construct the so-called fractional Gamma noise Gel’fand triple.
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References
Atangana, A.: Fractal–fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 102(2017), 396–406 (2017)
Ahokposi, D.P., Atangana, A., Vermeulen, D.P.: Modelling groundwater fractal ow with fractional differentiation via Mittag–Leffler law. Eur. Phys. J. Plus 132, 165–175 (2017)
Asai, N., Kubo, I., Kuo, H.: Multiplicative renormalization and generating functions I. Taiwan. J. Math. 7(1), 89–101 (2003)
Albeverio, S., Daletsky, Y.L., Kondratiev, Y.G., Streit, L.: Non-Gaussian infinite-dimensional analysis. J. Funct. Anal. 138, 311–350 (1996)
Brouers, F.: The fractal (BSf) kinetics equation and its approximations. J. Mod. Phys. 5, 1594–1601 (2014)
Bock, W., Desmettre, S., da Silva, J.L.: Integral representation of generalized grey Brownian motion. Stoch. Int. J. Probab. Stoch. Process. 92(4), 552–565 (2020)
Barhoumi, A., Ouerdiane, H., Riahi, A.: Pascal white noise calculus. Stoch. Int. J. Probab. Stoch. Process. 81(N3–4), 323–343 (2009)
Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications, Probability and Its Applications. Springer, London (2007)
Chen, W., Liang, Y.: New methodologies in fractional and fractal derivatives modeling. Chaos Solitons Fractals 102, 72–77 (2017)
Daletsky, Y.L.: Biorthogonal analogue of Hermite polynomials and the inversion of the Fourier transform with respect to a non-Gaussian measure. Funct. Anal. Appl. 25(2), 138–140 (1991)
Grothaus, M., Jahnert, F., Riemann, F., da Silva, J.L.: Mittag–Leffler analysis I: construction and characterization. J. Funct. Anal. 268, 1876–1903 (2015)
Hida, T.: Analysis of Brownian Functional. Carleton Mathematics Lecture Notes No. 13. Carleton University, Ottawa (1975)
Laskin, N.: Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003)
Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics. Springer, Berlin (2008)
Mainardi, F., Gorenflo, R., Scalas, E.: A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 53–64 (2004)
Mainardi, F., Mura, A., Pagnini, G.: The M-Wright function in time-fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 2010, 1–29 (2010)
Kondratiev, Y.G., Lytvynov, E.W.: Operators of Gamma white noise calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Topics 03(03), 303–335 (2000)
Kondratiev, Y.G., da Silva, J.L., Streit, L.: Generalized Appell systems. Methods Funct. Anal. Topol. 3(3), 28–61 (1997)
Riahi, A., Rebei, H.: Pascal white noise harmonic analysis on configuration spaces and applications. Infin. Dimens. Anal. Quantum Probab. Relat. Topics 21(04), 1850024 (2018)
Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)
Shaoa, W.-K., He, Y., Panb, J.: Some identities for the generalized Laguerre polynomials. J. Nonlinear Sci. Appl. 9, 3388–3396 (2016)
Pollard, H.: The complete monotonic character of the Mittag–Leffler function \(E_{\alpha }(-x)\). Bull. Am. Math. Soc. 54, 1115–1116 (1948)
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The authors extend their appreciation to King Saud University in Riyadh, Saudi Arabia for funding this research work through Researchers Supporting Project Number (RSPD2024R683).
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Mohamed Ayadi: Formal analysis, investigation, resources, writing, review and editing. Anis Riahi: Data curation, methodology, writing—review and editing. Mohamed Rhaima: Methodology, project administration, writing—review and editing. Hamza Ghoudi: Data curation, methodology, writing—review and editing.
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Communicated by Palle Jorgensen.
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This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory” edited by Marek Bozejko, Palle Jorgensen and Yuri Kondratiev.
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Ayadi, M., Riahi, A., Rhaima, M. et al. Fractional Gamma Noise Functionals. Complex Anal. Oper. Theory 18, 92 (2024). https://doi.org/10.1007/s11785-024-01534-0
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DOI: https://doi.org/10.1007/s11785-024-01534-0