Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure onto $\mathcal{S}'(\mathbb{R})$. Its characteristic functional is defined, for $f\in \mathcal{S}(\mathbb{R})$, by $$\hat{\mu}(f) = \int_{\mathcal{S}'(\mathbb{R})} \mathrm{e}^{\mathrm{i} \langle u ,f\rangle} \mathrm{d}\mu (u).$$

There are strong connexions between the support of the measure $\mu$ and the continuity of the characteristic functional of $\mu$. For instance, if $\mu$ is the measure associated with the Gaussian white noise with variance $1$, we have $\hat{\mu}(f) = \exp(-\frac{1}{2}\lVert f\rVert_2^2)$, that is continuous over $L_2(\mathbb{R})$. Applying for instance Theorem A.2 of [1], we deduce that $$\mathrm{Support}(\mu) \subset W_2^{-1/2-\epsilon}(\mathbb{R})$$ for every $\epsilon>0$. This requires only to remark that the identity is an Hilbert-Schmidt operator from $W_2^{1/2+\epsilon}(\mathbb{R})$ to $L_2(\mathbb{R})$.

Question: Is there a similar result if we know that the characteristic functional $\hat{\mu}$ of $\mu$ is continuous over $L^p$ for $1 \leq p < 2$?

[1] T. Hida and Si Si, An Innovation Approach to Random Fields

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability measure onto $\mathcal{S}'(\mathbb{R})$. Its characteristic functional is defined, for $f\in \mathcal{S}(\mathbb{R})$, by $$\widehat{\mathscr{P}}(f) = \int_{\mathcal{S}'(\mathbb{R})} \mathrm{e}^{\mathrm{i} \langle u ,f\rangle} \mathrm{d}\mathscr{P} (u).$$

There are strong connexions between the support of the measure $\mathscr{P}$ and the continuity of the characteristic functional of $\mathscr{P}$. For instance, if $\mathscr{P}$ is the measure associated with the Gaussian white noise with variance $1$, we have $\widehat{\mathscr{P}}(f) = \exp(-\frac{1}{2}\lVert f\rVert_2^2)$, that is continuous over $L_2(\mathbb{R})$. Applying for instance Theorem A.2 of [1], we deduce that $$\mathscr{P}\left( W_2^{-1/2-\epsilon}(\mathbb{R};(1+\lvert \cdot \rvert^2)^{(-1/2-\epsilon)/2})\right) =1$$ for every $\epsilon>0$, with $W_2^{s}(\mathbb{R};(1+\lvert \cdot \rvert^2)^{r/2})$ the weighted Sobolev space with regularity $s$ and weight function $(1+\lvert \cdot \rvert^2)^{r/2}$. This means that the Gaussian white noise is located in $W_2^{-1/2-\epsilon}(\mathbb{R};(1+\lvert \cdot \rvert^2)^{(-1/2-\epsilon)/2})$. This requires only to remark that the identity is an Hilbert-Schmidt operator from $W_2^{1/2+\epsilon}(\mathbb{R};(1+\lvert \cdot \rvert^2)^{(1/2+\epsilon)/2})$ to $L_2(\mathbb{R})$.

Question: Is there a similar result if we know that the characteristic functional $\widehat{\mathscr{P}}$ of $\mathscr{P}$ is continuous over $L^p$ for $1 \leq p < 2$?

NB. This is motivated by the fact that the characteristic functional of a S$\alpha$S white noise, of the form $\widehat{\mathscr{P}}(f) = \exp ( - \gamma^\alpha \lVert f \rVert^{\alpha}_{\alpha} )$, is continuous over $L^{\alpha}(\mathbb{R})$.

[1] T. Hida and Si Si, An Innovation Approach to Random Fields