# On a real smooth version of white noise distribution theory

In white noise analysis, one starts with a real Gelfand triple $$\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$$ and produces out of it, using complexifications along the way, the complex spaces of white noise test functions $$(\mathcal{N})$$ and white noise distributions $$(\mathcal{N})^{*}$$. The elements of $$(\mathcal{N})$$ are functions on $$\mathcal{N}^{*}$$ which are characterized by the analyticity and growth properties of their extensions to the complexification $$\mathcal{N}_{\mathbb{C}}^{*}$$.

My question is: Is there a smooth (or real analytic) version of white noise analysis developed in which the white noise test functions are characterized by their smoothness and growth on $$\mathcal{N}^{*}$$, rather than on $$\mathcal{N}_{\mathbb{C}}^{*}$$?

My motivation is as follows. I would like to work with a larger space of white noise test functions which contains all smooth (or real analytic) bounded real valued functions on $$\mathcal{N}^{*}$$. This is certainly not the case for the standard construction of $$(\mathcal{N})$$ because there are analytic functions on $$\mathcal{N}_{\mathbb{C}}^{*}$$ growing rapidly so that they do not belong to $$(\mathcal{N})$$ but their restrictions to $$\mathcal{N}^{*}$$ are bounded. Take, for example, $$F(\cdot)=e^{-\langle \cdot, f \rangle^n}$$, where $$f \in \mathcal{N}$$ and $$n$$ is even.

• Can you clarify... In which context is it typical to pass to the complexification? In the ones I'm familiar with (e.g. statistical physics & QFT), the complexification isn't used. For example, Minlos' theorem works just fine with real spaces. May 9 at 19:41
• @user1504, In all expositions of white noise analysis I'm familiar with, complex analyticity is used to characterize the white noise test functions and distributions and no real versions are given. For example, see these books: google.com/books/edition/White_Noise/…
– S.Z.
May 9 at 21:51
• – S.Z.
May 9 at 21:53
• – S.Z.
May 9 at 21:55
• @user1504, google.com/books/edition/… (the most explicit characterization of white noise test functions I know of is in this book, Chapter IV, section 2.3)
– S.Z.
May 9 at 21:58