A note on the Lévy Laplacian (Q1337364)

Let \(F\) be an element of Hida distributions \((S)^*\). Define \[ \partial^ 2_ i F = I_ 2(e_ i \otimes e_ i) F - I_ 2 (e_ i \otimes e_ i) : F - 2 \partial^*_ i \partial_ iF, \] where \(\partial_ i\) is the derivation with respect to the \(i\)-th base of a c.o.n.s. of basic space \(L^ 2\), \(I_ 2 (e_ i \otimes e_ i) (x) = \langle x, e_ i \rangle^ 2 - 1\) and \(:\) is the Wick product. Put \(\varphi_ n = {1 \over n} \sum_{i = 0}^{n - 1} I_ 2 (e_ i \otimes e_ i)\). Then the Lévy Laplacian \(\Delta_ LF = \lim_{n \to \infty} \sum_{i = 0}^{n - 1} \partial^ 2_ iF\) has a new expression \(\Delta_ LF = \lim_{n \to \infty} \varphi_ n F\), and it holds \(\Delta_ L (\varphi : \psi) = \varphi : \Delta \psi + \psi : \Delta_ L \varphi\).