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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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