Let $\xi$ be a random variable valued in the space of Schwartz distributions $\mathcal{S}'(\mathbb{R}^d)$.

For any open set $R\subset\mathbb{R}^d$ let $\Sigma(R)$ be the $\sigma$-algebra generated by $\{\xi(f)\,|\,f\in\mathcal{S}(\mathbb{R}^d), \,\textrm{supp}\,f\subset R\}$.

For any $\lambda\in(0,\infty)$ the random variable $\xi_\lambda$ valued in $\mathcal{S}'(\mathbb{R}^d)$ is defined by $\xi_\lambda(f) = \xi(f_\lambda)$ for all $f\in\mathcal{S}(\mathbb{R}^d)$, where $f_\lambda(x) = \lambda^{d/2}\,f(\lambda x)$.

Assume that:

(1) the $\sigma$-algebras $\Sigma(R_1)$ and $\Sigma(R_2)$ are independent for any disjoint open sets $R_1,R_2\subset\mathbb{R}^d$.

(2) for any $\lambda\in(0,\infty)$ the law of $\xi_\lambda$ coincides with the law of $\xi$.

Is it true that under the above assumptions $\xi$ must be the white noise?