Generalized functionals of a \(p\)-adic white noise (Q1363246)

In connection with \(p\)-adic probability theory the theory of a white noise with \(p\)-adic values is developed. The first author [Sov. Phys., Dokl. 37, No. 2, 81-83 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 322, No. 6, 1075-1079 (1992; Zbl 0768.60002) and Theor. Math. Phys. 97, No. 3, 1340-1348 (1993); translation from Teor. Mat. Fiz. 97, No. 3, 348-363 (1993; Zbl 0839.60005)] considered a frequency approach analogous to R. von Mises' theory; in [Russ. Acad. Sci., Dokl., Math. 46, No. 2, 373-377 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 5, 796-800 (1992; Zbl 0789.60001)] he proposed axiomatics of the \(Q_p\)-valued probability theory analogous to \textit{A. N. Kolmogorov}'s axiomatics [``Foundations of the theory of probability'' (1956)] where the non-Archimedean-valued measure theory was used. The formalism set forth can be compared with the calculus of generalized functionals of a white noise. The foundations of this calculus were laid out by T. Hida. We use the formalism of the Gaussian integration in the \(Q_p\)-valued case, which was proposed by the first author [Russ. Math. Surv. 45, No. 4, 87-125 (1990); translation from Usp. Mat. Nauk 45, No. 4(274), 79-110 (1990; Zbl 0722.46040)]. The generalized Brownian functionals of a \(p\)-adic white noise serve as a mathematical basis for many models of \(p\)-adic mathematical physics [see, e.g., \textit{V. S. Vladimirov}, \textit{I. V. Volovich} and {\textit{E. I. Zelenov}, ``\(p\)-adic analysis and mathematical physics'' (1994; Zbl 0864.60048) for \(p\)-adic physics]. These are, in the first place, analogs of the Dirichlet forms over \(Q_p\), as well as the Feynman integration. The questions of physical applications are considered in more detail by the authors [``\(p\)-adic valued white noise functionals'' (Preprint, Dep. Math., Huazhong Univ. of Sci. and Technol., Wuhan, 1993)].}