Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d).$
We define for all $r\geq 0,f\in L^2(\mathbb{R}_+ \times \mathbb{R}^d),Y_r(f)=\xi((q,x) \to 1_{[0,r]}(q)p_{r-q}*f(x)),$ $p_r(x)=(4\pi r)^{-d/2}e^{-|x|^2/(4r)}.$
Can we find a processes $U_r \in \mathcal{E}'$ such that for all $r,f\in \mathcal{E},U_r(f)=Y_r(f)$ a.s. ($\mathcal{E}$ is the space of Schwartz functions)?
