Elements of Lévy analysis on the spaces of nonregular test and generalized functions

• Datum: 2025
• Hauptverfasser: Kachanovsky, N., Качановський, Микола
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2025
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/8488
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.98 We present a survey of some author's results related to the development of the  L\'evy analysis on the spaces of nonregular test and generalized functions.  The indicated spaces are constructed by using Lytvynov's generalization of the chaotic representation property, which is a direct analog of decomposition of square-integrable random variables in the Hermite orthogonal polynomials in the Gaussian analysis. In this approach, numerous  definitions and statements are quite similar to their prototypes from the Gaussian analysis, which is very convenient for applications. The survey covers a fairly broad range of  issues, namely, an extended stochastic integral, a Hida stochastic derivative and their generalizations, the operators of stochastic differentiations and their analogs (generalizations), elements of the Wick  calculus, the relationship between the Wick calculus and integration, etc.