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I am starting to read about the Hida White Noise theory, and honestly there are some definitions/concepts that are not totally clear to me.

Start by assuming that $S(\mathbb R)$ is the space of rapidly decreasing functions and $S'(\mathbb R)$ denotes its dual.

It's easy to see that $S\subset L^2\subset S'$.

Then we define the "White noise probability space" as the triple $(S',\mathcal B(S'),\mu)$ where $\mathcal B$ stands for the Borel sigma algebra and $\mu$ is a generalized Gaussian measure (obtained by applying the Bochner-Minlos theorem).

At this point I would like to obtain two spaces such that the following holds

$$V\subset L^2(S')\subset V'.$$

This is basically what Hida did with his test function and distribution spaces, but I've seen many ways to construct them and I have some doubts regarding a particular one.

In this set of notes the author introduces this spaces by firstly defining

$(S)_p=\{\phi\in (L^2):\|\phi\|_p<\infty\}$ where $$\|\phi\|_p=\left(\sum_{n=0}^{\infty} \|(A^p)^{\otimes n} f_n\|_{L^2(\mathbb R^n)}^2\right)^{1/2}, p\geq 0.$$ Where we used the chaotic representation $\phi=\sum_n I_n(f_n), f_n\in \hat{L}^2(\mathbb R^n)$ and the differential operator is defined as $A= (-\Delta +x^2+1)$.

Now my question is, what's the justification for using this norm? From where does this differential operator come from?

I understand that (when defining the space of test functions) we need to build a "stronger" norm, but I don't see how we arrive to this particular construction of the norm $\|\cdot\|_p$.

Hope everything is clear and thanks in advance for any help.

Chaos
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1 Answers1

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As I recalled, e.g., in

https://mathoverflow.net/questions/366983/known-dense-subset-of-schwartz-like-space-and-c-c-infty/368043#368043

The Hermite functions $h_n$ form an orthonormal basis of $L^2(\mathbb{R})$ and also an unconditional Schauder basis of Schwartz space $\mathscr{S}(\mathbb{R})$. It follows that the map from temperate distributions to sequences of at most polynomial growth $\mathscr{S}(\mathbb{R})\rightarrow \mathscr{s}'(\mathbb{N})$, $T\mapsto (T(h_n))_{n\in\mathbb{N}}$ is an isomorphism of topological vector spaces. The $h_n$ are eigenvectors for the Hamiltonian of the harmonic oscillator, which essentially is $A$. This is why $A$ is used in the norm. If you use the TVS isomorphism to toss the space $\mathscr{S}$ in the trash and instead work with $\mathscr{s}'$ throughout your white noise investigations, the norm will look rather simple and natural.

Abdelmalek Abdesselam
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  • Thank you so much for you answer, there are a few concepts you mentioned that I don't quite understand, do you know of some references for this kind of stuff? – Chaos Aug 19 '20 at 20:47
  • You could start with the article by Barry Simon https://aip.scitation.org/doi/10.1063/1.1665472 but the proof of the TVS isomorphism there isn't great. He did a much better expository job in his new comprehensive treatise on analysis. – Abdelmalek Abdesselam Aug 19 '20 at 22:54
  • Hey Abdelmalek sorry for bothering you again, I've read the article you recommended me and also the Reed & Simon's Book and I think I understand it. Nonetheless when on Oksendal et al. book on PSDEs they introduce a norm that looks like $sum_{\alpha} c_{\alpha}(\alpha!)(2\mathbb N)^{k\alpha}$ where $\alpha$ is a multi-index and $c_{\alpha}$ is the coefficient in the Hermite chaos expansion. Do you know how is this related to the norms we've talked about before? – Chaos Aug 20 '20 at 13:35
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    I don't know about Oksendal et al. My guess is their system of seminorms indexed by $k$ is equivalent (defines the same LCTVS structure) as the Kuo norms indexed by $p$ translated in the language of sequences (coordinates in Hermite basis). I suppose there is also an additional translation to do between symmetric tensors $T_{i_1,\ldots,i_n}$ (needed for Kuo) and coefficients $c_{\alpha}$ (needed for Oksendal) where $\alpha_1$ is the number of $i$'s equal to one, etc. – Abdelmalek Abdesselam Aug 20 '20 at 14:41
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    Definitions for LCTVS equivalence are recalled here https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Aug 20 '20 at 14:43
  • thank you so much! I see it's not a trivial matter, I'll spend the next days trying to understand the generalities and how different constructions relate to each other. – Chaos Aug 21 '20 at 07:04