## Lévy Laplacian of generalized functions on a nuclear space.(English)Zbl 0749.46029

Summary: The Lévy Laplacian $\Delta F(\xi)=\lim_{X\to\infty} N^{- 1}\sum_{n=1}^ N\langle F''(\xi),e_ n\otimes e_ n\rangle$ is shown to be equal to
(i) $$\int_ T F_ s''(\xi;t)dt$$, where $$F_ s''$$ is the singular part of $$F''$$, and
(ii) $$2\lim_{\rho\to 0}\rho^{-2}(MF(\xi,\rho)-F(\xi))$$, where $$MF$$ is the spherical mean of $$F$$.
It is proved that regular polynomials are $$\Delta$$-harmonic and possess the mean value property. A relation between the Lévy Laplacian $$\Delta$$ and the Gross Laplacian $\Delta_ G F(\xi)=\sum_{n=1}^ \infty\langle F''(\xi),e_ n\otimes e_ n\rangle$ is obtained. An application to white noise calculus is discussed.

### MSC:

 46F25 Distributions on infinite-dimensional spaces 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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### References:

 [1] Gross, L., Potential theory on Hilbert space, J. Funct. Anal., 1, 123-181 (1967) · Zbl 0165.16403 [2] Hida, T., Analysis of Brownian Functionals, Carleton Mathematics Lecture Notes, Vol. 13 (1978), 2nd ed. · Zbl 0392.60055 [3] Hida, T., Generalized multiple Wiener integrals, (Proc. Japan Acad. Ser. A, 54 (1978)), 55-58 · Zbl 0389.60026 [4] Hida, T., Brownian Motion (1980), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0432.60002 [5] Hida, T.; Saitô, K., White noise analysis and the Lévy Laplacian, (Albeverio, S.; etal., Stochastic Processes in Physics and Engineering (1988), Reidel: Reidel Dordrecht), 177-184 [6] Kubo, I.; Takenaka, S., Calculus on Gaussian white noise, I, (Proc. Japan Acad. Ser. A, 56 (1980)), 376-380 · Zbl 0459.60068 [7] Kubo, I.; Takenaka, S., Calculus on Gaussian white noise, II, (Proc. Japan Acad. Ser. A, 56 (1980)), 411-416 · Zbl 0475.60064 [8] Kuo, H.-H, Gaussian Measures in Banach Spaces, (Lecture Notes in Mathematics, Vol. 463 (1975), Springer-Verlag: Springer-Verlag Berlin/New York) [9] Kuo, H.-H, Brownian functionals and applications, Acta Appl. Math., 1, 175-188 (1983) · Zbl 0527.60036 [10] Kuo, H.-H, Infinite Dimensional Stochastic Analysis, Nagoya University Lecture Notes (1984) [11] Kuo, H.-H, On Laplacian operators of generalized Brownian functionals, (Itô, K.; Hida, T., Stochastic Processes and Their Applications. Stochastic Processes and Their Applications, Lecture Notes in Mathematics, Vol. 1203 (1986), Springer-Verlag: Springer-Verlag Berlin/New York), 119-128 [12] Kuo, H.-H, The Fourier transform in white noise calculus (1989), preprint · Zbl 0686.60005 [13] Lévy, P., Leçons d’analyse fonctionnelle (1922), Gauthier-Villars: Gauthier-Villars Paris · JFM 48.0453.01 [14] Lévy, P., Problèmes concrets d’analyse fonctionnelle (1951), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0043.32302 [15] Obata, N., A note on certain permutation groups in the infinite dimensional rotation groups, Nagoya Math. J., 109, 91-107 (1988) · Zbl 0611.60013 [16] Obata, N., Analysis of the Lévy Laplacian, Soochow J. Math., 14, 105-109 (1988) · Zbl 0659.60104 [17] Obata, N., The Lévy Laplacian and mean value theorem, (Heyer, H., Probability Measures on Groups, IX. Probability Measures on Groups, IX, Lecture Notes in Mathematics, Vol. 1379 (1989), Springer-Verlag: Springer-Verlag Berlin/New York), 242-253 [18] Obata, N., A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J., 118, 111-132 (1990) · Zbl 0682.22015 [19] Polishchuk, E. M., Continual Means and Boundary Value Problems in Function Spaces (1988), Birkhäuser: Birkhäuser Basel/Boston/Berlin · Zbl 0685.31001 [20] Saitô, K., Itô’s formula and Lévy’s Laplacian, Nagoya Math. J., 108, 67-76 (1987) · Zbl 0646.60070 [21] Treves, F., Topological Vector Spaces, Distributions and Kernels (1967), Academic Press: Academic Press New York/London · Zbl 0171.10402
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