Based on the topological dual space \({\mathcal{F}}_{\theta }^{*}\left({\mathcal{S}{\prime}}_{\mathbb{C}}\right)\) of the space of entire functions with θ-exponential growth of finite type, we introduce a generalized stochastic Bernoulli–Wick differential equation (or a stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in \({\mathcal{F}}_{\theta }^{*}\left({\mathcal{S}{\prime}}_{\mathbb{C}}\right)\). This equation is an infinite-dimensional analog of the classical Bernoulli differential equation for stochastic distributions. This stochastic differential equation is solved and exemplified by several examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. H. Altoum, A. Ettaieb, and H. Rguigui, “Generalized Bernoulli–Wick differential equation,” Infinite Dimens. Anal. Quantum Probab. Relat. Top., 24, Issue 01, Article 2150008 (2021).
M. Ben Chrouda, M. El Oued, and H. Ouerdiane, “Convolution calculus and application to stochastic differential equation,” Soochow J. Math., 28, 375–388 (2002).
F. Cipriano, H. Ouerdiane, J. L. Silva, and R. Vilela Mendes, “A nonlinear stochastic equation of convolution type: solution and stochastic representation,” Global J. Pure Appl. Math., 4, No. 1, 2008.
R. Gannoun, R. Hachaichi, P. Krée, and H. Ouerdiane, “Division de fonctions holomorphes a croissance θ-exponentielle,” Technical Report E 00-01-04, BiBoS Univ. Bielefeld (2000).
R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgi, “Un théorème de dualité entre espace de fonction holomorphes à croissance exponentielle,” J. Funct. Anal., 171, 1–14 (2000).
T. Hida and N. Ikeda, “Analysis on Hilbert space with reproducing kernels arising from multiple Wiener integrals,” Proc. Fifth Berkeley Symp. Math. Stat. Prob., 2, part 1, 117–143 (1965).
H.-H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton (1996).
N. Obata, “White noise calculus and Fock spaces,” Lecture Notes Math., 1577, Springer-Verlag (1994).
N. Obata and H. Ouerdiane, “A note on convolution operators in white noise calculus,” Infinite Dimens. Anal. Quantum Probab. Relat. Top., 14, 661–674 (2011).
H. Rguigui, “Quantum λ-potentials associated to quantum Ornstein–Uhlenbeck semigroups,” Chaos Solitons Fractals, 73, 80–89 (2015).
H. Rguigui, “Characterization of the QWN-conservation operator,” Chaos Solitons Fractals, 84, 41–48 (2016).
H. Rguigui, “Characterization theorems for the quantum white noise gross Laplacian and applications,” Complex Anal. Oper. Theory, 12, 1637–1656 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 8, pp. 1085–1095, August, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i8.7223.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rguigui, H. Stochastic Bernoulli Equation on the Algebra of Generalized Functions. Ukr Math J 75, 1242–1254 (2024). https://doi.org/10.1007/s11253-023-02258-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-023-02258-8