For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed element $\omega \in C^\infty(\mathbb{T}^3, \mathbb{R})$ I constructed a Gaussian measure $\mu_\omega$ on $C^\infty(\mathbb{T}^3, \mathbb{R})$ whose characteristic functional is given by \begin{equation} \widehat{\mu_\omega}(T)=e^{\frac{1}{2}\int_{\mathbb{T}^3} (T*w)(x) [\Delta (T*w)](x) dx} \end{equation} where $T$ is an arbitrary continuous linear functional on $C^\infty(\mathbb{T}^3)$. In fact, it is known that $T$ can be identified with a periodic distribution on $\mathbb{R}^3$.
Now, let $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$ be a continuous map (NOT necessarily linear) such that \begin{equation} \int_{C^\infty(\mathbb{T}^3, \mathbb{R})} \lVert S(u) \rVert d\mu_{\omega}(u) < \infty \end{equation}
Then, the functional integration \begin{equation} \overline{S}:=\int_{C^\infty(\mathbb{T}^3, \mathbb{R})} S(u) d\mu_{\omega}(u) \end{equation} exists in the Hilbert space $L^2(\mathbb{T}^3, \mathbb{R})$.
Now, my question is as follows:
How "regular" is $\overline{S}$ as a periodic function on $\mathbb{R}^3$? Is it possible to establish continuity or even smoothness of $\overline{S}(x)$ with respect to $x \in \mathbb{R}^3$?
This kind of question seems somewhat familiar to me, but I cannot really find a relevant reference. Could anyone please help me?
